outer automorphism groups and bimodule categories of type ... · ainsi, la premi`ere personne que...

KATHOLIEKE UNIVERSITEIT LEUVEN

Faculteit Wetenschappen

Departement Wiskunde

Outer automorphism groups andbimodule categories of type II1

factors

Sebastien Falguieres

Promotor :

Prof. dr. Stefaan Vaes

Proefschrift ingediend tot het

behalen van de graad van

Doctor in de Wetenschappen

2009

Dankwoord

Ainsi s’acheve ma these et avec elle mes trois annees belges. Je me sou-viens tres bien du jour ou Stefaan m’annoncait que je pouvais partir fairema these en Belgique sous sa direction ; c’etait il y a deja trois ans, justeapres la soutenance de mon memoire de DEA. C’est justement au coursde cette annee de DEA que j’ai commence a m’interesser aux Algebresde von Neumann. La possibilite que m’offrait Stefaan de continuer a tra-vailler dans ce domaine etait alors une formidable opportunite.

Ainsi, la premiere personne que je tiens a remercier c’est toi Stefaan.Meme si avec le temps ne restent que les bons souvenirs, quitter la Franceet avec elle, famille et amis n’a pas ete chose facile au debut. Heureuse-ment tu m’as tout de suite oriente vers des problemes mathematiques mo-tivants et tu as toujours su etre disponible. C’est grace aux nombreusesidees et conseils avises que tu m’as souvent donnes que j’ai pu mener abien mes travaux. En effet, nos discussions, mathematiques ou non, ontete frequentes et m’ont toujours redonne le courage et la motivation qui acertains moments de la these viennent a manquer. Ce manuscrit tel qu’ilest aujourd’hui, resumant mon travail de these, doit beaucoup a la relec-ture minutieuse que tu en as faite et je t’en remercie. Je suis fier d’avoirete ton eleve.

Je tiens maintenant a remercier Claire Anantharaman, Dietmar Bisch etJohan Quaegebeur pour tous leurs commentaires pertinents concernantune version preliminaire de cette these. Enfin, je suis reconnaissant a RafCluckers, Mark Fannes et Alain Valette d’avoir accepte de faire partie demon jury.

Je souhaite remercier chaleureusement tous mes amis francais qui m’ontsoutenu depuis mon depart. Je pense notamment a David et Laam, maisaussi a Matthieu mon compagnon cycliste ainsi qu’a Fred et Caro. Jepense aussi a Cyril avec qui je suis reste en contact constant meme depuisla Californie et avec qui j’ai toujours eu plaisir a discuter. Son enthou-siasme communicatif pour les mathematiques qu’il pratique m’a souventredonne confiance. Enfin, je n’oublie pas Frederic et Omar avec qui j’aietudie a Paris mais aussi, Aude et Gerald et bien evidemment Carine.Merci a vous tous !

I also want to address a special thank to Anselm, Reiji and Yoshiko forall the fun time we had when they lived in Leuven. I always took benefit

from the mathematical discussions that I had with Reiji and I am verygrateful to him for sharing with me his knowledge about minimal actions,this was a precious help to start writing Chapter 7 ! Also many thanksto all my friends from the mathematics department at K.U.Leuven and aspecial word for my officemate Steven, the only person I know speakingfluently so many computer languages !

Je remercie egalement Claire Anantharaman, Jean Renault et Eric Ricardde m’avoir invite a venir exposer mes travaux dans leurs seminaires. I amalso grateful to Yasuyuki Kawahigashi for inviting me for a long stay atTokyo university. I discovered there a very pleasant place to work and agreat city !

Je conclurai en remerciant bien evidemment toute ma famille pour m’avoirtoujours accompagne et soutenu durant ces annees d’etudes.

Sebastien Falguieres.Leuven, le 22 juin 2009.

Contents

Introduction 1

I Preliminaries 15

1 An introduction tovon Neumann algebras 17

1.1 Bounded operators on a Hilbert space . . . . . . . . . . . 17

1.2 von Neumann algebras . . . . . . . . . . . . . . . . . . . . 19

1.3 Murray and von Neumann’s classification . . . . . . . . . 21

1.4 Gelfand-Naimark-Segal construction . . . . . . . . . . . . 25

1.5 Group von Neumann algebras . . . . . . . . . . . . . . . . 26

1.6 Murray and von Neumann’s group measure space construc-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.7 von Neumann algebras and ergodic theory . . . . . . . . . 31

2 Some topics in von Neumann algebra theory 33

2.1 Normalizers and Quasi-normalizers . . . . . . . . . . . . . 33

2.2 Group von Neumann algebras twisted by a 2-cocycle . . . 34

2.3 Property (T) and relative property (T) for II1 factors . . 38

2.3.1 Bimodules over von Neumann algebras . . . . . . . 38

2.3.2 Property (T) . . . . . . . . . . . . . . . . . . . . . 39

2.4 Outer actions of countable groups on II1 factors . . . . . . 41

ii CONTENTS

2.5 The ∗-algebra of operators affiliated with a II1 factor . . . 44

II The category of bimodules over a type II1 factor 47

3 An introduction to bimodules 49

3.1 Direct sums of modules. . . . . . . . . . . . . . . . . . . . 50

3.2 Connes’ tensor product of bimodules . . . . . . . . . . . . 51

3.3 Contragredient bimodule . . . . . . . . . . . . . . . . . . . 58

3.4 Intertwiners . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Coupling constant, Jones index 63

4.1 Dimension of modules over a II1 factor . . . . . . . . . . . 63

4.2 The Jones index of a subfactor . . . . . . . . . . . . . . . 66

4.3 Basic construction . . . . . . . . . . . . . . . . . . . . . . 68

5 Finite index bimodules 71

5.1 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Connes’ tensor product . . . . . . . . . . . . . . . . . . . 74

5.2.1 More on bounded vectors, Pimsner-Popa basis . . 74

5.2.2 Basic formulas . . . . . . . . . . . . . . . . . . . . 78

5.3 Intertwiners . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.1 General facts . . . . . . . . . . . . . . . . . . . . . 79

5.3.2 Connes tensor product versus product in a givenmodule . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.3 Intertwining bimodules . . . . . . . . . . . . . . . 84

5.4 Decomposition into irreducibles . . . . . . . . . . . . . . . 88

5.5 Contragredient bimodule . . . . . . . . . . . . . . . . . . . 89

5.6 Conjugates and Frobenius reciprocity . . . . . . . . . . . . 89

5.7 Dimension function . . . . . . . . . . . . . . . . . . . . . . 93

CONTENTS iii

6 Fusion algebras and bimodule categories 101

6.1 The fusion algebra of a finite index bimodule . . . . . . . 101

6.1.1 Quasi-normalizers and bimodules . . . . . . . . . . 103

6.1.2 Outer automorphism groups, fundamental groupsand fusion algebras . . . . . . . . . . . . . . . . . . 105

6.1.3 Outer actions of countable groups and fusion algebras106

6.1.4 Fusion algebra of almost normalizing bimodules. . 110

6.1.5 Freeness and free products of fusion algebras . . . 113

6.2 C∗-tensor categories and finite index bimodules . . . . . . 115

7 Minimal actions of compact groups 119

7.1 Representation theory for compact groups: some basicsnotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2 Spectral subspaces . . . . . . . . . . . . . . . . . . . . . . 122

7.2.1 General facts . . . . . . . . . . . . . . . . . . . . . 123

7.2.2 MG-MG-bimodules . . . . . . . . . . . . . . . . . . 125

7.3 Minimal actions and bimodule categories . . . . . . . . . . 130

7.4 A biduality result . . . . . . . . . . . . . . . . . . . . . . . 145

III Computations of outer automorphism groups andbimodules categories of amalgamated free product II1

factors 149

8 Amalgamated free product II1 factors 151

8.1 Definition and basic properties . . . . . . . . . . . . . . . 151

8.2 Some rigidity results: work of Ioana, Peterson, Popa . . . 158

9 Every compact group is the outer automorphism group ofa II1 factor 171

9.1 Notations and preliminary lemmas . . . . . . . . . . . . . 174

9.2 A Kurosh automorphism theorem for fusion algebras . . . 177

iv CONTENTS

9.3 Proof of Theorem 9.1 and Corollary 9.2 . . . . . . . . . . 179

9.4 Alternative proof of the main theorem . . . . . . . . . . . 184

10 The representation category of any compact group is thebimodule category of a II1 factor 185

10.1 Minimal actions and bimodule categories . . . . . . . . . . 187

10.2 Proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . 189

10.3 Proof of Theorem 10.2 . . . . . . . . . . . . . . . . . . . . 214

A Functor Rep(G) → Bimod(P ) 221

B Frobenius reciprocity theorem 225

Bibliography 226

Index of keywords 231

Index of symbols 234

Introduction

Von Neumann algebras are unital ∗-subalgebras of the algebra of boundedoperators on a Hilbert space that are closed under the weak operatortopology. They are characterized, by von Neumann’s bicommutant theo-rem, as being the unital ∗-subalgebras of the algebra of bounded operatorson a Hilbert space that are equal to their bicommutant. Murray and vonNeumann classified these algebras in three types. They proved in [58]that classifying von Neumann algebras amounts to classifying those withtrivial center. The von Neumann algebras for which the center only con-sists of scalar multiples of the identity and which are thus the most non-commutative ones, are called factors and are classified in types I, II andIII. Factors are the von Neumann algebras that cannot be decomposedinto the direct sum of two other von Neumann algebras.

In this thesis, we focus on factors of type II1. These factors are infinitedimensional factors endowed with a faithful normal trace. We call a tracenormal if its restriction to the unit ball is weakly continuous. Considernow a countable group Γ such that the conjugacy classes hgh−1 | h ∈ Γare infinite for all g 6= e. Such groups with infinite conjugacy classes(called ICC groups) appear frequently. Indeed, the free groups Fn, theinfinite symmetric group S∞ :=

⋃n≥1 Sn and the groups PSL(n,Z), with

n ≥ 2, are all ICC. The von Neumann algebra generated by the left regularrepresentation of the ICC countable group Γ yields a type II1 factor L(Γ),called the group von Neumann algebra of Γ. Although this constructionis very natural, distinguishing between group factors is usually very hardand many problems about group von Neumann algebras are open. Forexample, it is still unknown whether the group von Neumann algebrasL(F2) and L(F3) are isomorphic or not. In order to distinguish betweenII1 factors, several invariants were introduced, as we explain now.

2 Introduction

Murray and von Neumann defined the fundamental group F(M) of theII1 factor M as the following subgroup of R∗

+

F(M) := t > 0 |M t ∼= M .

In the case where t ≤ 1, the amplification M t is defined by M t := pMp,where p is a projection of trace t in M ; see section 6.1.2 for the precisedefinition. It can be proven that F(M) is a multiplicative subgroup ofR∗

+. This group has a fascinating history. Murray and von Neumannat their time were only able to prove that the fundamental group ofL(S∞) is the whole of R∗

+. One had to wait forty years, until the work ofConnes to obtain the existence of II1 factors for which the fundamentalgroup is different from R∗

+. In [7] Connes proved that the fundamentalgroup of the group factor of an ICC property (T) group is countable.However, even if the fundamental group of L(SL(3,Z)) follows countableby Connes’ result, there are still no computations of it, even nowadays.Voiculescu proved in [56] that the fundamental group of L(F∞) containsQ+ and Radulescu proved in [45] that it is the whole of R∗

+. Althoughit is still unknown if the the factors L(Fn) and L(Fm) are isomorphicor not, for n 6= m Radulescu in [44] and Dykema in [14] proved thefollowing alternative. Either all L(Fn) are isomorphic and in that casetheir fundamental group is the whole of R∗

+, either they are two by two nonisomorphic and their fundamental group is trivial. The first examples ofII1 factors with trivial fundamental group were given by Popa in [36, 37].He proved that L

(SL(2,Z) n Z2

)has trivial fundamental group, thus

answering a long standing open problem. He went even further in theunderstanding of fundamental groups since he proved in [34] that everycountable subgroup of R∗

+ can arise as the fundamental group of a II1factor. Some other constructions of factors with prescribed countablefundamental group were also obtained later by Ioana, Peterson, Popa [21]and Houdayer [20]. Very recently, Popa and Vaes [42, 41] proved that thefundamental group actually ranges over a large family of uncountablesubgroups of R∗

+, which was also an open problem for many years.

The outer automorphism group of M , defined by

Out(M) :=Aut(M)Inn(M)

is another important invariant of M . The outer automorphism groupof the hyperfinite II1 factor R is a huge group, it contains for example

3

every locally compact second countable group and every countable groupacts via outer automorphisms on R. The outer automorphism group of aII1 factor is also hard to compute in general. In [7] Connes proved thatthe outer automorphism group of the group factor of an ICC property(T) group is countable. Then, one had to wait until the work of Ioana,Peterson and Popa in 2004 to obtain really new results. They answereda long standing open problem, proving the existence of II1 factors forwhich every automorphism is inner. They proved, more generally, thatevery abelian compact group arises as the outer automorphism group ofa II1 factor. These results were still only existence results. But later,Popa and Vaes [43] gave explicit examples of II1 factors with trivial outerautomorphism group. They even proved that every finitely presentedgroup can be explicitly realized as the outer automorphism group of aII1 factor. Popa and Vaes also gave examples of ICC groups Γ for whichOut(L(Γ)) ∼= Char(Γ) o Out(Γ). In his ten problem list [23] Jones hadasked whether this isomorphism could hold for arbitrary ICC property(T) groups and this is still unknown. These results were refined later byVaes [53] where he proved that every countable group can arise as theouter automorphism group of a II1 factor.

One of the richest invariants of a II1 factor M is the bimodule categoryBimod(M) consisting of all M -M -bimodules MHM of finite Jones index:dim(MH) < ∞ and dim(HM ) < ∞. See Chapter 4 for the definition ofthe Jones index. Equipped with the Connes tensor product, Bimod(M)is a C∗-tensor category; see Chapter 3 and Chapter 6 for these notions.The category Bimod(M) completely encodes both the fundamental groupF(M) and the outer automorphism group Out(M) (see Proposition 6.5).Furthermore, Bimod(M) also encodes, in a certain sense, all subfactorsM0 ⊂M of finite Jones index [24]: performing Jones’ basic construction,we get M0 ⊂ M ⊂ M1 and obtain the M -M -bimodule ML2(M1)M; seesection 4.3 concerning the basic construction.

As a result, it seemed until recently quite hopeless to explicitly computeBimod(M) for any II1 factor M . Vaes obtained in [54] the existence ofII1 factors with trivial bimodule category and hence also trivial subfac-tor structure, trivial fundamental group and trivial outer automorphismgroup. This result was still an existence result, following the work of [21].The first concrete II1 factors with trivial bimodule category were alsoobtained by Vaes in [53] which includes as well concrete examples of II1

4 Introduction

factors where Bimod(P ) is a Hecke-like category.

Description of the chapters and statement of the

main results

This thesis is divided in three parts. The first part is an introductoryone. In the first chapter we give a general and basic introduction to vonNeumann algebras. We also mention some key results that were obtainedin this field. We do not try to provide an exhaustive list of the recentresults, we would rather highlight some important connections with otherareas of mathematics such as infinite group theory and ergodic theory. Inthe second chapter we go deeper into the theory of von Neumann algebrasand focus on specific topics that will form the general background materialfor this thesis such as Popa’s relative property (T) for von Neumannalgebras, outer actions of countable groups on II1 factors or the algebraof operators affiliated with a II1 factor.

The second part of the thesis is entirely devoted to bimodules over II1factors and consists of five chapters. This part does not contain newresults and is included in order to make the text rather self-contained.Chapter 3 is an introduction to bimodules. The material gathered inthis chapter is taken from Bisch’s paper [1]. In Chapter 4, we explainhow modules (not bimodules) over a II1 factors are completely classifiedby a number between 0 and +∞ called dimension or coupling constant,already introduced by Murray and von Neumann in [30]. We give basicproperties of the coupling constant and the Jones index of an inclusionof II1 factors. In Chapter 5, we investigate properties of bimodules withfinite left and right coupling constant. These bimodules are called finiteindex bimodules. It will become clear throughout the text that finite indexbimodules are much richer, and also more difficult to analyze, than the leftor right modules. This chapter is meant to be a toolbox providing concretelemmas to work with bimodules. At the end of the chapter, we recall theconstruction of a dimension function on the class of finite index bimodules,using the language of C∗-tensor categories with conjugates. In Chapter 6,finite index bimodules over II1 factors are analyzed in a more global way.It will be clear, after the foregoing chapters, that the class of finite indexbimodules over II1 factors, modulo unitary equivalence, has a naturalstructure of fusion algebra. In this chapter, we also recall some usual

5

notions and constructions associated to fusion algebras such as freenessor free product of fusion algebras. We also prove that the fundamentalgroup and the outer automorphism group of M are entirely encoded byfinite index M -M -bimodules. In the last chapter of this second partwe study minimal actions of compact groups on II1 factors. We recallthe construction of a fully faithful tensor functor between the categoryRep(G) of finite dimensional representations of the compact group G andthe category Bimod(MG) of finite index MG-MG-bimodules, where MG

denotes the fixed-point algebra. This result was obtained in the 80’s (see[47]), and formulated in the context of algebraic quantum field theory.

The third part of the thesis contains the results I obtained during myPh.D research. I provide computations of the outer automorphism groupand the bimodule category of certain type II1 factors. All these factorsare amalgamated free product von Neumann algebras. In Chapter 8, Irecall the construction and some basic properties of amalgamated freeproduct von Neumann algebras. I also state some important rigidity re-sults for amalgamated free product von Neumann algebras obtained byIoana, Peterson and Popa in [21]. This chapter also recalls a few essentialresults from [54] where Vaes proves the existence of II1 factors with trivialbimodule category. For the convenience of the reader, I give a proof ofmost of the statements. In Chapter 9 and Chapter 10 I give a detailedversion of the papers [15, 16]. These two papers were written in collabora-tion with Stefaan Vaes. I prove the existence of type II1 factors for whichthe outer automorphism group can be any prescribed second countablecompact group. I also prove the existence of type II1 factors with bimod-ule category isomorphic to the representation category of any prescribedsecond countable compact group. The starting point of the whole of thework done in this thesis is the paper [21] by Ioana, Peterson and Popa inwhich they proved, among many other deep results, that the outer auto-morphism group of a type II1 factor could be any abelian compact group.As a consequence, they obtained the first examples of II1 factors havingonly inner automorphisms. Their results are existence results. Popa hasdeveloped a theory of deformation and rigidity for group actions and vonNeumann algebras. He extended the notion of relative property (T) forgroups to the von Neumann setting. An inclusion N ⊂M having relativeproperty (T) in the sense of Popa is called rigid, see section 2.3. One canprove that L(Λ) ⊂ L(Γ) is rigid if and only if the pair of groups Λ ⊂ Γ hasthe relative property (T). His deformation/rigidity techniques led to sev-

6 Introduction

eral breakthroughs in the theory of von Neumann algebras during the lastfew years. His results also found important applications in ergodic theory.In his strong rigidity theorem obtained in [35], he was able to deduce forthe first time the conjugacy of group actions out of their von Neumannequivalence; see section 1.7 for a precise statement. The same defor-mation and rigidity techniques are the core of all the above mentionedresults [36, 37, 34, 21, 20, 42, 41, 53] on the fundamental group and theouter automorphism group of II1 factors. Amalgamated free products vonNeumann algebras allow lots of deformations. Ioana, Peterson and Popaapplied the deformation/rigidity techniques in this context and provedstriking rigidity results : they proved, for example, that any rigid vonNeumann subalgebra Q ⊂M0 ∗N M1 must be “weakly contained” in M0

or M1, see Theorem 8.7 and Theorem 8.8. In the terminology of Popa,saying that Q is weakly contained in M0 inside M := M0 ∗N M1 meansthat there exists a Q-M0-subbimodule of L2(M) which is finitely gener-ated as right M0-module. In the “good cases”, this weak containmentleads to actual unitary conjugacy. This technique is known as Popa’sintertwining by bimodules technique; see Theorem 5.19. The previousresult locating rigid subalgebras of amalgamated free product von Neu-mann algebras was a crucial ingredient to prove the existence of II1 factorswith prescribed abelian compact outer automorphism group. The factorsthey constructed are of the form M0 ∗N M1, where N is hyperfinite andMi = N o Γi, for countable groups Γi acting outerly on N (see section2.4 for the notion of outer action). The details of the strategy they usedcan be found in section 8.2 and a precise statement of their theorem isgiven in Theorem 8.15. Note that the factor M they construct is in factthe crossed product of N by the group Γ0 ∗ Γ1. Their assumptions onM are of three different types. There are assumptions on the groups,assumptions on the actions and they also make a “freeness assumption”in the following sense. They suppose that Γ1, viewed as a subgroup ofOut(N) is free with respect to another countable subgroup of Out(N) sothat, roughly, Γ1 and the normalizer of Γ0 in Out(N) live far away fromeach other. They also suppose that N ⊂ N o Γ0 is rigid. Following theirconstructions, we could generalize their methods and prove that even non-abelian compact groups can arise as the outer automorphism group of aII1 factor. Consider a minimal action σ of G on the hyperfinite II1 factorM1 and denote N := MG

1 . Let Γ0 be a countable group acting by outerautomorphisms on N . Let M0 := N o Γ0 and define M := M0 ∗N M1.

7

The leg M0 is identical to Ioana, Peterson and Popa’s work. We assumethat M0 has property (T). We exploit the different behavior of M0 andM1 (one is property (T), the other is hyperfinite) to prove, using Ioana,Peterson and Popa’s results, that every automorphism α ∈ Aut(M) sat-isfies, up to unitary conjugacy in M , that α(N) = N and α(M0) = M0.Here again, our assumptions on the actions of G and Γ0 are linked by afreeness assumption. A careful use of this freeness assumption will forcethe automorphism α to preserve M1 globally and imply that α fixes Npointwise, all this up to unitary conjugacy in M . Then, by some classicaltheory of outer actions and minimal actions, α|M0

is of the form αω, forsome character ω of Γ, where αω(auγ) = ω(γ)uγ and α|M1

= σg, for someelement g ∈ G. The precise theorem is the following. We use the notationFAlg(N) to denote the collection of all finite index N -N -bimodules of theII1 factor N , modulo unitary conjugacy.

Theorem I.1 (Theorem 9.1). Let M1 be the hyperfinite II1 factor andG a compact group acting on M1. Denote N = MG

1 , the von Neumannalgebra of G-fixed points in M1. Let Γ be an ICC group acting on N .Denote M0 := N o Γ. Assume that

1. the action σ : G y M1 is minimal,

2. the action Γ y N is outer and M0 has the property (T),

3. the natural images of RepG → FAlg(N) and Aut(N ⊂ M0)restr−→

Out(N) ⊂ FAlg(N) inside the fusion algebra FAlg(N), are free inthe sense of Definition 6.14.

Then, the homomorphism

Char(Γ)×G→ Aut(M0 ∗NM1) : (ω, g) 7→ αω ∗ σg

induces an isomorphism Char(Γ)×G ∼= Out(M0 ∗NM1).

We also prove the following.

Corollary I.2 (Corollary 9.2). Let G be a compact, second countablegroup and σ : G y R a minimal action on the hyperfinite II1 factor R.Let Γ := SL(3,Z). Then there exists an outer action of Γ on the fixed

8 Introduction

point algebra RG, such that for M given as the amalgamated free productM = (RG o Γ) ∗

RGR, the natural homomorphism

G→ Aut(M) : g 7→ id ∗ σg

induces an isomorphism G ∼= Out(M).

We deduce the previous corollary from Theorem I.1 above and anothercrucial ingredient that we explain now. We make use of the fact thatthe fusion algebra of the hyperfinite II1 factor is huge, in the sense thatit contains many free fusion subalgebras. Because of [54, Theorem 5.1](see Theorem 6.15), given any countable fusion subalgebras F0 and F1

of FAlg(R) there always exists, via a Baire category argument, an auto-morphism α of the hyperfinite II1 factor such that F0 “conjugated” byα becomes free with respect to F1. This theorem is a key point for ourproofs and is responsible of the fact that we only obtain existence results:we make two fusion subalgebras free, via a Baire category result. Ioana,Peterson and Popa’s theorem also relies on a similar Baire category typeresult (see [21, Lemma 1.2]) based on the fact that Out(R) contains manyfree countable subgroups. In both cases, for [54, Theorem 5.1] and [21,Lemma 1.2], the key ingredients come from [38]. This kind of argumentis also used in [54] where Vaes proves the existence of factors with trivialbimodule category.

In [16], we managed to prove a much stronger result. We prove the exis-tence of a II1 factor (M, τ) and minimal action of a given compact groupG on M such that writing P := MG, every finite index P -P -bimoduleis isomorphic with PMor(L2(M),Hπ)P for a uniquely determined finitedimensional unitary representation π : G→ U(Hπ). Here, the space

Mor(L2(M),Hπ) :=T : L2(M) → Hπ | T (σg(ξ)) = π(g)T (ξ),

for all ξ ∈ L2(M), g ∈ G

endowed with the scalar product 〈S, T 〉 := τ(S∗T ) is a P -P -bimodule, forthe following left and right P -actions:

(a·S·b)(ξ) := S(a∗ξb∗), for all S ∈ Mor(L2(M),Hπ), a, b ∈ P, ξ ∈ L2(M).

More precisely, we prove that

Rep(G) → Bimod(P ) : π 7→ PMor(L2(M),Hπ)P

9

defines an equivalence of C∗-tensor categories.

The proof is considerably more involved than our first results. We wantto understand all finite index bimodules, not only the ones for which theleft and right dimension is one (i.e the automorphisms). Still, our factorsof the form MG, where M is again an amalgamated free product over thehyperfinite II1 factor. We have to impose many conditions to be able tocontrol all such finite index MG-MG-bimodules. We do not give here allthe technical assumptions and refer to section 10.2 for more information.However we make the following remarks. Let G be a second countablecompact group acting on the hyperfinite II1 factor M1. Denote N = MG

1

and fix an inclusion N ⊂ M0 into the II1 factor M0. We are interestedin the II1 factor M := M0 ∗N M1 and extend the action G y M1 to anaction G y M by acting trivially on M0. We denote by P := MG thefixed point II1 factor. We will assume that M0 contains a subfactor N0

with property (T). The first step of our proof relies on a careful analysis ofall finite indexM0-P -bimodules. As for the analysis of the automorphismsin Theorem I.1, we start with the “property (T) leg M0” and use hereagain the rigidity results of Ioana, Peterson and Popa.

We also make the following construction. Consider the group

Γ := (Q3 ⊕Q3) o SL(3,Q) ,

defined by the action A · (x, y) = (Ax, (At)−1y) of SL(3,Q) on Q3 ⊕ Q3.Let Λ := Z3 ⊕ Z3. Choose an irrational number α ∈ R and define the2-cocycle Ω ∈ Z2(Γ, S1) such that

Ω((x, y), (x′, y′)

)= exp

(iα(〈x, y′〉 − 〈y, x′〉)

), for all (x, y),

(x′, y′) ∈ Q3 ⊕Q3 ,

Ω(g,A) = Ω(A, g) = Ω(A,B) = 1, for all g ∈ Γ ,

A,B ∈ SL(3,Q) .

Theorem I.3 (Theorem 10.1). Let G be a compact, second countablegroup. Define N := LΩ(Λ) and M0 := LΩ(Γ) with the groups Λ,Γ andthe cocycle Ω as above. Let M1 be the hyperfinite II1 factor. DenoteM := M0 ∗N M1. Then, there exists a minimal action G y M such thatRep(G) → Bimod(MG) is an equivalence of C∗-tensor categories.

Once again, this theorem uses [54, Theorem 5.1], leading again to anexistence result.

Some notations and

conventions

Generalities

We use the Kronecker symbol on the set I defined by

δi,j : I → 0, 1 : k 7→

0 if i 6= j

1 if i = j

Let A ⊂ H be a subset of the Hilbert space H. We denote by [A] theclosed vector space generated by A.

We will always denote by Tr the non-normalized trace on Mn(C) and bytr the normalized one.

Hilbert spaces and operators

All our inner products 〈·, ·〉 are linear in the second variable. The normassociated is denoted by ‖ · ‖.All Hilbert spaces are assumed to be separable.

We denote by ei ∈ Cn the natural vectors.

The algebraic tensor product is denoted by . The tensor product of theHilbert spaces H0 and H1 is denoted by H0 ⊗H1.

We denote by B(H,K) the vector space of bounded operators from H toK. We denote by B(H) := B(H,H) for the algebra of bounded operatorson H.

If ξ ∈ H, we denote by ξ∗ : H → C : η 7→ 〈ξ, η〉. We denote by H thedual Hilbert space H := ξ∗ | ξ ∈ H.

12 Introduction

We write Cn(Cm)∗ instead of B(Cm,Cn). We implicitly consider Cn(Cm)∗

as a Hilbert space with scalar product 〈ξ, η〉 = Tr(ξ∗η). Obviously,Cn(Cm)∗ is an Mn(C)-Mm(C)-bimodule.

We use the leg numbering notation defined as follows. Let H1,H2,H3 beHilbert spaces. Denote by Σ the flip map on H2⊗H3. Then we have thefollowing bounded operators on H1 ⊗H2 ⊗H3.

• T ∈ B(H1⊗H3) =⇒ T13 := (1⊗Σ∗)(T⊗1)(1⊗Σ) ∈ B(H1⊗H2⊗H3),

• T ∈ B(H1 ⊗H2) =⇒ T12 := T ⊗ 1 ∈ B(H1 ⊗H2 ⊗H3),

• T ∈ B(H2 ⊗H3) =⇒ T23 := 1⊗ T ∈ B(H1 ⊗H2 ⊗H3).

von Neumann algebras

All von Neumann algebras are assumed to have separable predual. Wedenote by M∗ the predual of M .

For any von Neumann algebra M we denote Mn := Mn(C)⊗M .

Let M be a von Neumann algebra. We say that elements (eij) ∈M forma system of matrix units if they satisfy the following.

• eijekl = δj,keil .

• e∗ij = eji.

•∑

i eii = 1.

All ∗-homomorphisms between von Neumann algebras are implicitly as-sumed to be normal.

We denote by (M, τ) the von Neumann algebra endowed with the faithfulnormal trace τ . We also call M a finite von Neumann algebra or a tracialvon Neumann algebra.

Let (M, τM ) be a tracial von Neumann algebra. Unless explicit mention,all inclusions of von Neumann algebras (N, τN ) ⊂ (M, τM ) are assumedto satisfy

• τM |N = τN ,

• 1N = 1M .

13

Let M0 and M1 be von Neumann algebras. The algebraic tensor productof M0 and M1 is denoted by M0 M1. The tensor product of M0 andM1 is denoted by M0⊗M1.

Part I

Preliminaries

Chapter 1

An introduction to


1.1 Bounded operators on a Hilbert space

Let (H, 〈·, ·〉) be Hilbert space. We denote by ‖ · ‖ the norm given by theinner product 〈·, ·〉. The linear maps T : H → H are continuous wheneverthey are bounded and the (operator) norm of T is defined by

‖T‖ := infM > 0 | ‖T (ξ)‖ ≤M‖ξ‖, for all ξ ∈ H .

Such maps are called bounded operators on H. The collection of allbounded operators on H, denoted by B(H), is a unital algebra. Thealgebra B(H) is complete for the operator norm which satisfies in partic-ular

‖ST‖ ≤ ‖S‖‖T‖, for all S, T ∈ B(H) .

An algebra with such properties is called a Banach algebra . Furthermore,B(H) is endowed with an operation

∗ : B(H) → B(H) : T 7→ T ∗

defined by〈Tξ, η〉 = 〈ξ, T ∗η〉, for all ξ, η ∈ H .

The operator T ∗ is called the adjoint of T . The ∗-operation is 2-periodic,anti-linear, anti-multiplicative and norm-preserving. An application with

18Chapter 1. An introduction to


such properties is called an involution. An element x ∈ B(H) such thatx = x∗ is called selfadjoint. An element x ∈ B(H) such that x∗x = 1 =xx∗ is called a unitary. The unitary elements form a group denoted U(H).Note that an element x ∈ B(H) is an isometry (i.e. x preserves the innerproduct) if and only if x∗x = 1. We also have that

‖T ∗T‖ = ‖T‖2, for every T ∈ B(H) .

A Banach algebra endowed with an involution satisfying the foregoingequality is called a C∗-algebra. In fact, any unital C∗-algebra is a normclosed unital ∗-subalgebra of B(H) for some Hilbert space H. The algebraof continuous functions on a compact space is a unital C∗-algebra andactually, every commutative C∗-algebra arises in this way. The algebraB(H) can be endowed with many other topologies that are weaker thanthe one inherited from the norm.

We define some weak topologies on B(H). The following list is not ex-haustive.

• The strong topology is the topology generated by the family ofsemi-norms (x 7→ ‖xξ‖)ξ∈H . So,

xi → x iff ‖(xi − x)ξ‖ → 0, for all ξ ∈ H .

• The weak topology is the topology generated by the family of semi-norms (x 7→ |〈xξ, η〉|)ξ,η∈H . So,

xi → x iff |〈(xi − x)ξ, η〉| → 0, for all ξ, η ∈ H .

• The ultraweak topology is the topology generated by the family ofsemi-norms

(x 7→

∣∣∑n∈N〈xξn, ηn〉

∣∣)(ξn),(ηn)∈`2(N)⊗H . So,

xi → x iff

∣∣∣∣∣∑n∈N

〈(xi − x)ξn, ηn〉

∣∣∣∣∣→ 0, for all (ξn), (ηn) ∈ `2(N)⊗H .

• The ultrastrong topology is the topology generated by the familyof semi-norms

(x 7→

∑n∈N ‖xξn‖2

)(ξn)∈`2(N)⊗H . So,

xi → x iff∑n∈N

‖(xi − x)ξn‖2 → 0, for all (ξn) ∈ `2(N)⊗H .

1.2 von Neumann algebras 19

Since |〈xξ, η〉| ≤ ‖xξ‖‖η‖ ≤ ‖x‖‖ξ‖‖η‖, the weak topology is weakerthan the strong one, both being weaker than the norm topology. Theweak topology is weaker than the ultraweak, itself weaker than the ultra-strong. These topologies in general do not behave well with respect to thealgebraic structure. For example, the ∗-operation and the multiplicationmap (S, T ) 7→ ST are not strongly continuous. Furthermore, the strongand ultrastrong topologies coincide on the unit ball of B(H). Similarly,the weak and ultraweak topologies coincide on the unit ball.

1.2 von Neumann algebras

From now on, we will focus on the study of unital ∗-subalgebras of B(H)that are closed under the weak operator topology defined above. Thesealgebras are called von Neumann algebras. If a = a∗ in the von Neumannalgebra M , the Borel functional calculus of a belongs to M as well. Thisvery important fact allows to do lots of computations inside the von Neu-mann algebra. Note that B(H) itself is a von Neumann algebra. It can beproven that every abelian von Neumann algebra is of the form L∞(X,µ),where (X,µ) denotes a standard probability space. This first definitionof von Neumann algebras is purely analytic but von Neumann algebrascan be defined in two other equivalent ways as we explain now.

Von Neumann algebras can be characterized in a purely algebraic manner,by von Neumann’s bicommutant theorem. For every non-empty subsetM ⊂ B(H), one defines its commutant M ′ by

M ′ := T ∈ B(H) | xT = Tx, for all x ∈M .

The commutant of a von Neumann algebra M is still a von Neumannalgebra.

Theorem 1.1. Let M ⊂ B(H) be a unital ∗-subalgebra, where 1M is theidentity operator on B(H). Then, the following are equivalent.

• M is a von Neumann algebra.

• M = (M ′)′.

Von Neumann algebras can also be characterized without referring to anyrepresentation on a Hilbert space. Indeed, Sakai proved that the space



M∗, called the predual of M , and consisting of ultraweakly continuouslinear functionals on M is the unique Banach space such that M = (M∗)∗.Von Neumann algebras are the C∗-algebras that are the dual of a Banachspace. By construction of M∗, the ultraweak topology on M coincideswith the weak∗-topology on M , viewed as the dual of M∗. This space-free characterization of a von Neumann algebra proves that the ultraweaktopology does not depend on the choice of the Hilbert space where thevon Neumann algebra is represented.

In fact, the ultraweakly continuous linear functionals on a von Neumannalgebra can be characterized by an algebraic property, called normality.Before giving this property, we make a little digression to talk aboutpositive elements in a C∗-algebra. We know that a matrix a is positive ifand only if a is selfadjoint with positive eigenvalues. There is an analogousnotion of positivity in arbitrary C∗-algebras. An element a of a unitalC∗-algebra is said to be positive if a is selfadjoint and

Sp(a) := λ ∈ C | a− λ1 is non-invertible ⊂ R+ .

The subset Sp(a) is called the spectrum of a. The positive elements of aC∗-algebra A form a convex cone that we denote A+. An element a of aC∗-algebra is positive and and only if it can be written a = b∗b, for someb ∈ A. The set of self-adjoints elements in A is endowed with the partialordering given by

a ≤ b if b− a ∈ A+ .

The following properties are often useful. Let a, b ∈ A be self-adjointelements.

• ‖a‖ ≤ k ⇐⇒ −k1 ≤ a ≤ k1,

• a ≤ b⇒ x∗ax ≤ x∗bx, for all x ∈ A.

If M is a von Neumann algebra, the positive cone M+ has the followingextra property. If (xi) is an increasing and bounded net in M+ (in thesense that supi ‖xi‖ is bounded), then, supxi ∈M+. We can now definethe notion of normal linear functional on M .

Definition 1.2. A linear functional ω : M → C on the von Neumannalgebra M is said to be

• positive if ω(x) ≥ 0, for every x ∈M+,

1.3 Murray and von Neumann’s classification 21

• normal if it is positive and ω(supxi) = supω(xi) for every increasingand bounded net (xi) ∈M .

Linear functional are very important for the analysis of von Neumannalgebras and among them, the normal ones are of particular interest to ussince they define the ultraweak topology. Indeed, it can be proven thata linear functional is normal if and only it is positive and ultraweaklycontinuous.

1.3 Murray and von Neumann’s classification

Murray and von Neumann classified von Neumann algebras in three types.They proved in [58] that classifying von Neumann algebras amounts toclassifying those for which the center Z(M) := M∩M ′ is trivial. The vonNeumann algebras for which the center only consists of scalar multiplesof the identity and which are thus the most non-commutative ones, arecalled factors and are classified in types I, II and III. Factors are theindecomposable von Neumann algebras, indeed, one can prove that M isa factor if and only if M is not the direct sum of two other von Neumannalgebras. The classification of factors relies on a careful study of thegeometry of the self-adjoint idempotents, called projections. Projectionsare crucial objects in von Neumann algebra theory, for example, the linearspan of all projections in the von Neumann algebra M is norm-dense inM .

Before giving more details concerning the classification of factors, wereview some important definitions and properties concerning projections.Projections are positive elements and we have p ≤ q if and only if Im p ⊂Im q. So we have that p ≤ q if and only if pq = p. The projections pand q are called orthogonal if Im p and Im q are orthogonal vector spaces.Note that p is orthogonal to q if and only if pq = 0. Then, it is easilyverified that the sum of orthogonal projections is still a projection. Theset of projections in a von Neumann algebra form a lattice because forevery family of projections (pi)i∈I ∈M , the projections∧

i∈Ipi, with range

⋂i∈I

Im pi



and ∨i∈I

pi, with range span⋃i∈I

Im pi

are respectively the largest projection in M smaller than all the pi andthe smallest projection in M majorizing all the pi.

Definition 1.3. Let M be a von Neumann algebra and x ∈M .

• The smallest projection p ∈ M such that px = x is called the leftsupport projection of x.

• The smallest projection p ∈M such that xp = x is called the rightsupport projection of x.

• The smallest central projection z such that x = zx is called thecentral support of x.

Let M ⊂ B(H) be a von Neumann algebra and x ∈M . Note that the left(resp. right) support of x is the projection on the closure of xH (resp.x∗H).

An element v ∈ B(H) is called a partial isometry if v is an isometry on aclosed subspace of H and identically zero on its orthogonal complement.Equivalently, v is a partial isometry if and only if v∗v is a projection,which is called the initial projection of v. In this case vv∗ is also aprojection, called the final projection of v. Every bounded operator a ona Hilbert space admits a polar decomposition given by a := u|a|, where|a| := (a∗a)1/2 and u is a partial isometry with initial projection equal tothe right support projection of a. An important fact is that the partialisometry coming from the polar decomposition of an element of a vonNeumann algebra M still belongs to M .

Definition 1.4. Let M be a von Neumann algebra and p, q projectionsin M .

• p and q are said to be equivalent, and we write p ∼ q if there existsa partial isometry v ∈M such that p = v∗v and q = vv∗.

• p is said to be sub-equivalent to q, and we write p . q if there existsp0 ≤ q such that p ∼ p0.

1.3 Murray and von Neumann’s classification 23

It is easy to verify that ∼ is an equivalence relation. We have the Cantor-Bernstein type result saying that if p . q and q . p then, p ∼ q. Havingequivalent projections in a von Neumann algebra is not an exceptionalphenomenon. Indeed, the partial isometry arising in the polar decompo-sition the element a of the von Neumann M entails the equivalence ofthe initial and final support projections of a. In a factor, two arbitraryprojections p, q can be compared: either p . q, either q . p.

A projection p in the von Neumann algebra M is called an atom (or aminimal projection ) if pMp = Cp, or if it satisfies, equivalently,

q ≤ p⇒ q = p or q = 0, for every projection q ∈M .

A von Neumann algebra is called diffuse if it contains no minimal projec-tions. In fact, a factor is atomic if and only if it is isomorphic to B(H),for some Hilbert space H, which may be finite dimensional. Those factorsare called type I factors. If M is an infinite dimensional type I factor, wesay that M is a type I∞ factor.

We say that a projection p in a von Neumann algebra M is finite if

p ∼ p0 ≤ p⇒ p0 = p .

A projection which is not finite is called infinite. So, an infinite projectionis equivalent to a strictly smaller sub-projection. This definition shouldbe compared with the definition of an infinite set, which is set in bijectivecorrespondence with a strictly smaller subset. Note that every minimalprojection is finite. A factor is said to be of type II if it has finite projec-tions but no minimal projections. A factor is of type III if every non-zeroprojection is infinite. Among type II factors, we have the following dis-tinction. A type II factor for which the identity is a finite projection iscalled a type II1 factor. A type II factor for which the identity is aninfinite projection is called a type II∞ factor.

Before giving an equivalent formulation of Murray and von Neumann’sclassification, we introduce some important terminology, constantly usedin the sequel.

Definition 1.5. Let M be a von Neumann algebra. A positive linearfunctional τ : M → C is called a state if it satisfies τ(1) = 1. A state τ iscalled trace if it satisfies τ(xy) = τ(yx), for all x, y ∈M .



We say that the positive linear functional ϕ on the von Neumann algebraM is faithful if ϕ(x∗x) = 0 implies that x = 0, for every x ∈ M . On afactor, the faithfulness of normal states is automatic. We also define thenotion of infinite traces. We already know that the linear functional

Tr : B(H)+ → [0,+∞] : T →∑i

〈Tei, ei〉 ,

where (ei) is an orthonormal basis of H, is a trace. The trace Tr onlytake finite values on trace-class operators. More generally, we have thefollowing definition.

Definition 1.6. An ultraweakly lower semi-continuous positive-linearfunctional Tr : M+ → [0,+∞] on the von Neumann algebra M satis-fying

Tr(x∗x) = Tr(xx∗), for all x ∈M ,

is called a normal trace. We say that the trace Tr is semi-finite if

x ∈M+ | Tr(x) < +∞

is ultraweakly dense in M+.

Then, we can state the following theorem, due to Murray and von Neu-mann.

Theorem 1.7. Let M be a factor. Then

• M is of type I if and only if M ∼= B(H) for some Hilbert space H.

• M is of type II1 if and only if M is infinite dimensional and admitsa normal tracial state.

• M is of type II∞ if and only if M admits a semi-finite trace Tr,such that Tr(1) = +∞ and M is not isomorphic to B(H) for someinfinite dimensional Hilbert space H.

• M is of type III if every normal trace is zero.

It can be proven that the trace on a II1 factor is necessarily unique.Murray and von Neumann started by constructing a dimension functionon the set of projections in a factor M , which is a map D : M → [0,+∞]such that

1.4 Gelfand-Naimark-Segal construction 25

1. p ∼ q ⇐⇒ D(p) = D(q),

2. p . q ⇐⇒ D(p) ≤ D(q),

3. D(p+ q) = D(p) +D(q), for orthogonal projections p and q,

4. D takes finite values on finite projections.

The dimension function they constructed is unique up to multiplicationby a scalar. Then, they proved in [29], that in the II1 case this dimensionfunction could be extended to a trace onM . The trace on a II1 factor givesa notion of continuous dimension in the sense that for every t ∈ [0, 1],there exists a projection p ∈ M such that τ(p) = t. From now on, wewill focus on type II1 factors. We give some classical constructions anddiscuss the problem of distinguishing between type II1 factors, which isin general a very hard task.

1.4 Gelfand-Naimark-Segal construction

By definition, von Neumann algebras are represented on a a Hilbert space.It is sometimes crucial to be able to construct another representation ofthe same von Neumann algebra on another Hilbert space. See for exampleChapter 8 where we construct amalgamated free product von Neumannalgebras: the important point of the construction is to build an appropri-ate representation. We recall here the Gelfand-Naimark-Segal construc-tion, called GNS construction, for a von Neumann (M, τ) endowed withthe faithful normal tracial state τ . This construction holds in a more gen-eral setting, see for example [52, Theorem 9.14]. We refer to the section 9of the first chapter of [52] for the proofs of all the results in this section.

Definition-Proposition 1.8. Let (M, τ) be a von Neumann algebra witha faithful normal finite trace τ . There exists a unique (up to unitaryequivalence) representation λτ : M → B(Hτ ) on the Hilbert space Hτ

with a vector ξτ satisfying

1. [λτ (M)ξτ ] = Hτ and

2. τ(a) = 〈ξτ , λτ (a)ξτ 〉



The Hilbert space Hτ is denoted L2(M, τ) or simply L2(M), when noconfusion is possible and is obtained as the completion of M with respectto the inner product given by

〈a, b〉 := τ(a∗b), for all a, b ∈M .

We denote by ‖ · ‖2 the norm inherited from this inner product. It iseasy to see that the left multiplication on M extends to the required∗-representation λτ .

Note that since τ is faithful, the representation λτ is also faithful so, wecan see M ⊂ B(L2(M)).

A very interesting fact is that M is a subset of the Hilbert space L2(M).

The map x 7→ x∗ extends to an anti-unitary operator JM : L2(M) →L2(M) and we have the anti-representation ρτ : M → B(L2(M)) given byρτ (x) := JMx

∗JM . It is easy to see that ρτ and λτ commute. Further-more, we have the following important relation

λτ (M)′ ∩ B(L2(M)) = ρτ (M) .

Example 1.9. Let Γ be an countable and ICC group. Since the vectorδe ∈ `2(Γ) is cyclic for the left regular representation λ : Γ → B(`2(Γ)),we have that L2(L(Γ)) = `2(Γ).

1.5 Group von Neumann algebras

Let Γ be a countable group. The left regular representation of λ : Γ →B(`2(Γ)) is defined by λg(δh) = δgh, where (δg) denotes the canonicalorthonormal basis of `2(Γ). The group von Neumann algebra L(Γ) of thegroup Γ is defined by

L(Γ) := λg | g ∈ Γ′′ .

This von Neumann algebra is endowed with the faithful normal tracegiven by τ(x) = 〈δe, xδe〉. The group von Neumann algebra L(Γ) is afactor (and thus of type II1) if and only if the conjugacy classes hgh−1 |h ∈ Γ of every element g 6= e are infinite. Groups with infinite conjugacyclasses are called ICC groups. Such groups appear frequently, for examplethe free groups Fn, the infinite symmetric group S∞ :=

⋃n≥1 Sn and

1.5 Group von Neumann algebras 27

the groups PSL(n,Z), with n ≥ 2, are all ICC. One can prove, usingthe Fourier transform, that for a countable abelian group Γ, the groupvon Neumann algebra L(Γ) is isomorphic with L∞(Char(Γ),Haar), whereChar(Γ) denotes the group of characters of Γ. So, for example, we havethat L(Z) ∼= L∞(S1).

The GNS space associated to L(Γ) is given by `2(Γ) since the vector δeis cyclic of L(Γ). Every element x ∈ L(Γ) can be written as the L2-convergent sum x =

∑g∈Γ xgug where xg ∈ C and the ug are unitaries

satisfying ugh = uguh. Writing elements of L(Γ) in such way is very usefulfor concrete computations.

Murray and von Neumann proved in [31] the existence of a unique (upto isomorphism) hyperfinite II1 factor, defined as the bicommutant ofan increasing union of matrix algebras. The hyperfinite finite II1 factor,always denoted by R, can be realized as the group von Neumann algebraL(S∞). Murray and von Neumann were able to distinguish between thetwo II1 factors L(S∞) and L(F2), using an invariant that they calledproperty (Γ). A II1 factor is said to have property (Γ) if it admits non-trivial central sequences (un) ∈ U(M) in the sense that ‖[x, un]‖2 tendsto zero, for every x ∈ M . We say that a central sequence (un) ∈ U(M)is trivial when ‖un − τ(un)1‖2 tends to zero. Murray and von Neumannproved that all central sequences of L(F2) are trivial while L(S∞) admitsnon-trivial central sequences. Distinguishing between group factors isusually very hard and many problems about group von Neumann algebrasremain open. For example, it is not known whether the group factorsof F2 and F3 are isomorphic or not. Due to a celebrated theorem ofConnes, rigidity phenomena deducing the isomorphism of the groups Γand Λ from the isomorphism of their group von Neumann algebras L(Γ)and L(Λ), should only be expected in the non-amenable setting. Weexplain this fact. A countable group is said to be amenable if the leftregular representation admits a sequence of almost invariant unit vectors.For example, the group Z but also all abelian and all solvable groupsare amenable. The amenability property can be translated in the vonNeumann setting as follows. It can be proven that the group Γ is amenableif and only if the von Neumann algebra L(Γ) is injective:

Definition 1.10. A von Neumann algebra M ⊂ B(H) is injective ifthere exists a conditional expectation of B(H) onto M i.e a linear mapEM : B(H) →M satisfying the following.



• EM is positive and EM (1) = 1

• EM (aTb) = aEM (T )b, for every a, b ∈M and T ∈ B(H).

The very deep result proven by Connes in [8] is that a II1 factor is injectiveif and only if it is hyperfinite. So, all the group von Neumann algebrasarising from ICC amenable groups are isomorphic.

One way of contradicting amenability is to have Kazhdan property (T)(see section 2.3). All SL(n,Z), with n ≥ 3 are property (T) groups.Connes and Jones extended the notion of property (T) to the von Neu-mann level in [9] (see section 2.3). They also proved that a property (T)II1 factor cannot be embedded into a free group factor. Connes conjec-tured in [6] that ICC property (T) groups are isomorphic if and only iftheir group von Neumann algebras are isomorphic and this conjecture iscompletely open. As a consequence of his strong rigidity theorem (seeTheorem 1.13) Popa proves that wreath-products of Z with ICC w-rigidgroups are distinguished by their group von Neumann algebras. We callwreath product of Z with Γ the group Z o Γ := (

⊕Γ Z) o Γ where Γ acts

on⊕

Γ Z via Bernoulli shift : g · (xh)h := (xg−1h)h. A group is calledw-rigid if it contains an infinite normal subgroup with relative property(T); see section 2.3. The group SL(2,Z) n Z2 is w-rigid.

1.6 Murray and von Neumann’s group measure

space construction

Let Γ be a countable group and M ⊂ B(H) a von Neumann algebra. Wesay that the group Γ acts on M , via σ, and we write Γ y M , if σ isa homomorphism Γ → Aut(M). Given such action, we construct a vonNeumann algebra, called crossed product , encoding at the same time thedata of the group Γ and the action Γ y M . We represent M on H⊗`2(Γ)via the representation π given by

π(a)(ξ ⊗ δg) := σg−1(a)ξ ⊗ δg, for all a ∈M, ξ ∈ H, g ∈ Γ .

We represent L(Γ) on H⊗`2(Γ) via id⊗λ, where λ denotes the left regularrepresentation of Γ:

(id⊗ λ)(h)(ξ ⊗ δg) := ξ ⊗ δhg, for all ξ ∈ H, g, h ∈ Γ .

1.6 Murray and von Neumann’s group measure spaceconstruction 29

The crossed product of M by Γ is the von Neumann algebra

M o Γ := π(M), (id⊗ λ)(Γ)′′ ⊂ B(H ⊗ `2(Γ)) .

We will denote by ug := 1 ⊗ λg the unitaries defining the action of Γ onH ⊗ `2(Γ) and we identify a ∈ M with π(a). Then, one can verify thefollowing commutation relation

ugau∗g = σg(a), for all a ∈M, g ∈ Γ . (1.1)

Note that in the case where M = C and σ is the trivial action, the crossedproductMoΓ is just the group von Neumann algebra L(Γ). Furthermore,if Γ is a countable group acting by automorphisms on the countable groupΛ, then one can prove that L(Λ o Γ) ∼= L(Λ) o Γ. For example, we havethat L(Z2 o SL(2,Z)) ∼= L(Z2) o SL(2,Z) ∼= L∞(S2) o SL(2,Z).

We are interested in the situations where MoΓ is a factor. We introducethe following terminology concerning actions of groups on von Neumannalgebras.

Definition 1.11. Let σ : Γ y M be an action of the countable group Γon the von Neumann algebra M .

• The action σ is said to be properly outer, if(ax = σg(x)a, for all x ∈

M)⇒ a = 0.

• The action σ is said to be ergodic, if x ∈M | σg(x) = x = C1.

It can be proven that Γ acts on M properly outerly if and only if M oΓ ∩M ′ = Z(M). If the action is also ergodic, then M o Γ is a factor. Insection 2.4 we study outer actions of countable groups on II1 factors, inthat case, the resulting crossed product is factorial.

Suppose that M is a tracial von Neumann algebra. The GNS spaceassociated toMoΓ is given by L2(M)⊗`2(Γ). So, in particular, L2(MoΓ)is the orthogonal direct sum of copies of L2(M) indexed by Γ given byL2(Mug). Every element x ∈M o Γ can be written as the L2-convergentsum x =

∑g∈Γ xgug, where xg ∈M .

Crossed products can also be built out of the action of a countable groupon a standard Borel measured space. Let Γ act on a standard Borelmeasured space (X,B, µ). We say that the action Γ y (X,µ) is



• essentially free if x | g · x = x is µ-negligible, for all g 6= e,

• ergodic if it is indecomposable in the sense that any Γ invariantsubspace of X is either of measure 0 or 1,

• non-singular if µ(B) = 0 ⇒ µ(g ·B) = 0, for all g ∈ Γ.

In this setting all statements only hold up to measure zero. Every count-able group Γ admits an essentially free, ergodic and non-singular actionon the standard probability space given by the Bernoulli shift actionΓ y

∏Γ(X,µ) where g · (xh)h = (xg−1h)h.

Let Γ y (X,µ). Then, Γ acts on L∞(X,µ) via (g · F )(x) := F (g−1 · x).So, we can build the crossed product L∞(X,µ) o Γ also known as Mur-ray and von Neumann’s group-measure-space construction, which alreadyappeared in [30]. One can check that Γ y (X,µ) essentially freely (resp.ergodically) if and only if Γ y L∞(X,µ) properly outerly (resp. ergod-ically). Then, every essentially free and ergodic action of the countablegroup Γ on (X,µ) yields a factor L∞(X,µ)oΓ. The following actions areall essentially free, ergodic and will give examples of factors of all types,by Theorem 1.12 due to Murray and von Neumann.

1. Let θ 6∈ Q and let Z y RZ via k · z := e2iπkθz.

2. Let θ 6∈ Q and let Z + θZ y R by translation.

3. Let Q∗ n Q y R via (p, q) · x := px+ q.

Theorem 1.12. Let Γ y (X,µ) be an essentially free, ergodic, non-singular action. Denote M := L∞(X,µ) o Γ. Then,

• M is a type I factor if and only if Γ acts transitively.

• M is a type II1 factor if and only if there exists a non-atomic Γ-invariant finite measure in the class of µ.

• M is a type II∞ factor if and only if there exists a non-atomicΓ-invariant infinite measure in the class of µ.

• M is a type III factor if and only if there exists no non-trivial Γ-invariant measure in the class of µ.

1.7 von Neumann algebras and ergodic theory 31

1.7 von Neumann algebras and ergodic theory

In this section (X,µ) and (Y, ν) denote standard non-atomic probabilityspaces and we only consider essentially free, ergodic and probability mea-sure preserving actions of countable groups. The II1 factor L∞(X) o Γ isa von Neumann algebra containing a copy of L∞(X) and a copy of L(Γ)satisfying the covariance relation (1.1). So, L∞(X) o Γ captures infor-mation concerning the group Γ and the action Γ y (X,µ). Two naturalquestions arise:

• What kind of isomorphism of actions gives rise to isomorphic crossedproduct II1 factors?

• Conversely, what does an isomorphism L∞(X,µ)oΓ ∼= L∞(Y, ν)oΛtell us concerning the groups and the actions?

It is proven in [49, 17] that a probability space isomorphism ∆ : (X,µ) →(Y, ν) extends to an isomorphism of the crossed products L∞(X,µ)oΓ ∼=L∞(Y, ν)oΛ if and only if ∆(Γ ·x) = Λ ·∆(x), for µ-almost every x ∈ X.In that case the the actions Γ y (X,µ) and Λ y (Y, ν) are called orbitequivalent. Note that the groups Γ and Λ may not be isomorphic fororbit-equivalent actions Γ y (X,µ) and Λ y (Y, ν). There is a strongerway of comparing actions which goes as follows.

We say that actions Γ y (X,µ) and Λ y (Y, ν) are conjugate if thereexists an isomorphism of probability spaces ∆ : (X,µ) → (Y, ν) and anisomorphism δ : Γ → Λ such that ∆(g · x) = δ(g) ·∆(x), for all g ∈ Γ andµ-almost every x ∈ X. Then, the map∑

g∈Γ

xgug 7→∑g∈Γ

∆∗(xδ(g))uδ(g)

extends to an isomorphism L∞(X,µ)oΓ ∼= L∞(Y, ν)oΛ sending L∞(X,µ)to L∞(Y, ν). In particular, conjugate actions are orbit equivalent.

Conjugacy implies orbit equivalence which in turn implies von Neumannequivalence (isomorphism of the crossed products). All converse resultsare called rigidity results. To deduce orbit equivalence from von Neu-mann equivalence one needs to have an isomorphism of crossed productssending L∞(X) to L∞(Y ). This phenomenon is usually unexpected: for



example Connes and Jones in [10] gave examples of von Neumann equiv-alent actions that are not orbit equivalent. Moreover, a lot of informationcan be lost in the passage to the orbit equivalence; indeed Dye [13] provedthat all free ergodic actions of infinite abelian groups are orbit equivalent.The celebrated theorem of Ornstein and Weiss [32] proves that in fact allfree ergodic actions of infinite amenable groups are orbit equivalent.

We end this section with the following breakthrough rigidity result ob-tained by Popa in 2004 (see [34, 35]). With this theorem, Popa deducesfor the first time and for a large class of groups, the conjugacy of actionsout of their von Neumann equivalence. To obtain this very powerful resultPopa developed a theory of deformation/rigidity which was the startingpoint for many other rigidity results concerning group actions and vonNeumann algebras.

Theorem 1.13 (Popa, Theorem 7.1 in [35]). Let Γ be an ICC groupacting on (X,µ) via Bernoulli shift. Let Λ y (Y, ν) be an essentiallyfree, ergodic measure preserving action of the w-rigid group Λ. If

L∞(X,µ) o Γ ∼= L∞(Y, ν) o Λ ,

then, the groups Γ and Λ are isomorphic and their actions are conjugate.

Chapter 2

Some topics in von

Neumann algebra theory

2.1 Normalizers and Quasi-normalizers

Let (M, τ) be a tracial von Neumann algebra and N ⊂M a von Neumannsubalgebra.

• The normalizer of N inside M is defined as:

NormM (N) := u ∈ U(M) | uNu∗ = N .

• The inclusion N ⊂M is called regular if NormM (N)′′ = M .

• The quasi-normalizer of N inside M is defined by

QNM (N) =a ∈M

∣∣∣ ∃a1, . . . , an, b1, . . . , bm ∈M such that

Na ⊂n∑i=1

aiNand aN ⊂m∑i=1

Nbi

. (2.1)

• The inclusion N ⊂M is called quasi-regular if QNM (N)′′ = M .

Remark that the quasi-normalizer of N ⊂ M is a unital ∗-subalgebra ofM containing N and we have

N ′ ∩M ⊂ span(NormM (N)

)⊂ QNM (N) .

34 Chapter 2. Some topics in von Neumann algebra theory

Let Γ be a group and Λ ⊂ Γ a subgroup.

• The commensurator of Λ ⊂ Γ is defined as

CommΓ(Λ) := g ∈ Γ | gΛg−1 ∩ Λ has finite index

in gΛg−1 and in Λ .

• The inclusion Λ ⊂ Γ is called almost normal if CommΓ(Λ) = Γ.

A typical example of an almost normal subgroup is SL(n,Z) ⊂ SL(n,Q).Remark that the inclusion L(Λ) ⊂ L(Γ) is

• regular if Λ is a normal subgroup of Γ,

• quasi-regular if the inclusion Λ ⊂ Γ is almost normal.

We conclude this section on quasi-normalizers with the following usefullemma; see [55, Lemma 6.5].

Lemma 2.1. Let Q ⊂M be an inclusion of finite von Neumann algebrasand p ∈ Q, a non-zero projection. Then

QNpMp(pQp)′′ = pQNM (Q)′′p .

2.2 Group von Neumann algebras twisted by a

2-cocycle

In this section, we recall definitions and basics facts concerning twistedgroup von Neumann algebras. We start with the definition of the 2-cohomology of a countable group with values in S1, the complex numbersof modulus 1. In this section, the group Γ always denotes a countablegroup.

Definition 2.2. The set Z2(Γ, S1) of S1-valued 2-cocycles on the groupΓ is defined by

Z2(Γ, S1) := Ω : Γ× Γ → S1 | Ω(g, h)Ω(gh, k) = Ω(g, hk)Ω(h, k)

for all g, h, k ∈ Γ .

2.2 Group von Neumann algebras twisted by a 2-cocycle 35

The set ∂2(Γ, S1) of S1-valued coboundaries on the group Γ is defined by

∂2(Γ, S1) := Ω ∈ Z2(Γ, S1) | ∃a : Γ → S1, such that for all g, h ∈ Γ

a(g)a(h) = Ω(g, h)a(gh) .

For every function a : G→ S1, the map defined by

(∂a)(g, h) := a(gh)a(g)a(h)

is a 2-cocycle with vales in S1, called the boundary of a. Note that a is acharacter if and only if ∂a = 1.

The set Z2(Γ, S1) is an abelian group and the set ∂2(Γ, S1) is a normalsubgroup. The 2-cohomology of Γ with values in S1 the group defined by

H2(Γ, S1) :=Z2(Γ, S1)∂2(Γ, S1)

.

All examples of 2-cocycles are built from projective representations as weexplain now.

• Let π : Γ → U(H) be a projective representation. Then, there existsa 2-cocycle Ωπ : Γ× Γ → S1 such that, for all g, h ∈ Γ, π(g)π(h) =Ωπ(g, h)π(gh). We sometimes refer to Ωπ as the obstruction cocycleof π.

• Let π : Γ → U(H) be a unitary representation and a : Γ → S1.Then, the formula ρ(g) := a(g)π(g) defines a projective representa-tion of Γ. The obstruction cocycle Ωρ is a coboundary for the mapa.

Let Γ be a countable group acting by automorphisms on the countablegroup Λ. The following lemma gives a technique to extend Γ-invariant2-cocycles on Λ into 2-cocycles on Γ n Λ.

Lemma 2.3. Let Γ,Λ be countable groups. Let σ : Γ y Λ, by automor-phisms. Let Ω ∈ Z2(Λ, S1) such that

Ω(σs(g), σs(h)

)= Ω(g, h), for all g, h ∈ Λ and s ∈ Γ .

Then,

Ω((s, g), (t, h)

):= Ω

(g, σs(h)

), for all g, h ∈ Λ and s ∈ Γ

defines a 2-cocycle Ω ∈ Z2(Γ n Λ, S1).


We omit the proof which is a straightforward computation.

Let Ω ∈ Z2(Γ, S1). The left Ω-regular representation λΩ : Γ → U(`2(Γ))defined by

λΩ(g)δh = Ω(g, h)δgh

is a projective representation of Γ with cocycle Ω. The twisted group vonNeumann algebra LΩ(Γ) is defined as the von Neumann algebra generatedby the λΩ(g) . This von Neumann algebra is not always a factor but isendowed with the normal tracial state

τ(x) := 〈δe, xδe〉, x ∈ LΩ(Γ) .

As for the group von Neumann algebra, we easily prove that LΩ(Γ) isa factor when Γ is ICC. The converse does not hold anymore but thefollowing proposition gives an easy way of building factorial LΩ(Γ).

Notation 2.4. We denote by Char(Γ) the group of characters of theabelian countable group Γ. If G is a compact group, we still denote byChar(G) for the subset of Irr(G) consisting of one-dimensional unitaryrepresentations of G. If G is a compact abelian group, then Char(G) isthe set of continuous group homomorphisms ω : G→ S1. We recall thatfor an abelian countable group Γ, the group Char(Γ) is compact abelianand we have the Pontryagin duality Char(Char(Γ)) ∼= Γ.

Proposition 2.5. Let Γ be an abelian countable group. Suppose thatπ : Γ → Char(Γ) is a homomorphism. Then:

π : Γ → Char(Γ) : π(h)(g) = π(g)(h)

is a homomorphism and the following hold.

• Ωπ(g, h) := π(g)(h) defines a 2-cocycle,

• Ωπ 6∈ ∂2(Γ, S1) if there exists g ∈ G such that π(g) 6= π(g) ,

• LΩπ(Γ) is the hyperfinite II1 factor if π(g) 6= π(g) for all g 6= e.

Proof. One can easily check that Ωπ defines a 2-cocycle. We prove thesecond assertion. Suppose that a(g)a(h) = Ωπ(g, h)a(gh) for some mapa : G → S1. Abelianness of Γ implies that Ωπ(g, h) = Ωπ(h, g) and thusπ(g)(h) = π(g)(h) .

2.2 Group von Neumann algebras twisted by a 2-cocycle 37

Take x in the center LΩπ(Γ) and write the L2-convergent sum x :=∑g agλΩπ(g). An immediate computation yields that

agΩ(h, g) = Ω(hgh−1, h)ahgh−1 for all g, h ∈ Γ.

Since Γ is abelian, we have that agΩπ(g, h) = agΩπ(h, g), for every g, h ∈Γ. Whenever g 6= e, π(g) 6= π(g) yields the existence of an element h ∈ Γsuch that Ωπ(g, h) 6= Ωπ(h, g). Then, ag = 0, for all g 6= e.

If there exists a non-trivial g ∈ Γ such that π(g) = π(g), we have thatΩπ(g, h) = Ωπ(h, g), for all h ∈ Γ. This equality and the fact that Γ isabelian easily imply that λΩπ(g) is a non-trivial element in the center ofLΩπ(Γ).

We need to get examples of injective morphisms π : Γ → Char(Γ) to pro-duce factorial twisted group von Neumann algebras. We first introducesome terminology.

Definition 2.6. Let Γ be an abelian countable group. A 2-cocycle Ω ∈Z2(Γ, S1) is called a bi-character when Ω(·, g) and Ω(g, ·) are elements ofthe group of characters Char(Γ) of Γ, for all g ∈ Γ.

Let Ω be a bi-character on an abelian group Γ. Then, by definition, themap

π : Γ → Char(Γ) : (h 7→ Ω(g, h))

is a homomorphism.

Definition 2.7. Let Ω be a bi-character on an abelian group Γ. Thecocycle Ω is called non-degenerate when the associated homomorphismπ : Γ → Char(Γ) is injective with dense range.

Non-degenerate bi-characters are very useful to produce concrete real-izations of the hyperfinite II1 factor as a twisted group von Neumannalgebra.

We end this section with a standard example. Let z ∈ S1 . We define thefollowing bi-character on Z2 :

Ωz

((x, y), (x′, y′)

):= zxy

′−yx′ .

Then, if z 6∈ exp(2iπQ), LΩz(Z2) is a II1 factor.


2.3 Property (T) and relative property (T) for

II1 factors

The notion of property (T) and relative property (T) for II1 factors usesthe concept of bimodule over II1 factors, also called Connes’ correspon-dences. See [5] and also Popa’s notes [40]. Bimodules are intensivelystudied in this thesis starting from Part II but we give the definition now.

2.3.1 Bimodules over von Neumann algebras

Definition 2.8. Let N,M ⊂ B(H) be von Neumann algebras. Let π :M → B(H) be a normal representation and π′ : N → B(H) a normalcontinuous anti-representation .

• The Hilbert space H is a left M -module, for the left action givenby

a · ξ := π(a)ξ .

• The Hilbert space H is a right N -module, for the right action givenby

ξ · a := π′(a)ξ .

• When π and π′ commute, the Hilbert space H is called an M -N -bimodule.

The fact that the representation π and the anti-representation π′ commutestates the associativity of the left and right actions:

a · (ξ · b) = (a · ξ) · b .

We recall the notion of opposite algebraMop associated to a von Neumannalgebra M . As vector spaces M and Mop are the same but, if we denoteby xo the element x ∈M viewed in Mop, the von Neumann algebra Mop

is equipped with the product given by xo · yo := (yx)o. So, an M -N -bimodule is also the data of commuting normal representations of M andNop.

Let (M, τ) be a tracial von Neumann algebra. The GNS constructionL2(M, τ) provides the easiest example of anM -M -bimodule, with left andright actions given by the representation λτ and the anti-representationρτ defined in section 1.4.

2.3 Property (T) and relative property (T) for II1 factors 39

2.3.2 Property (T)

We recall the notion of relative property (T) of Kazhdan-Margulis forcountable groups; see for example [11]. Let G be a countable group andπ : G → U(H) a unitary representation. A sequence of unit vectors(ξn) ∈ H is said to be almost invariant for π if it satisfies

‖π(g)ξn − ξn‖ → 0, for all g ∈ G .

Definition 2.9. A pair of countable groups H ⊂ G is said to have therelative property (T) every representation π : G → U(H) that admitsa sequence of almost invariant unit vectors has a non-zero H-invariantvector.

The pair (Zn ⊂ Zn o SL(n,Z)) is the main example of pairs with relativeproperty (T). In the case H = G, we say that the group G has property(T). The groups SL(n,Z) all have property (T), for n ≥ 3.

A countable group G is said to be amenable if the left regular repre-sentation on `2(G) admits a sequence of almost invariant unit vectors.Then, clearly, the finite groups are the only groups being at the sametime amenable and property (T).

The concept of property (T) was extended to the von Neumann algebrasetting by Connes and Jones in [9] and can be stated as follows.

Definition 2.10. A II1 factor (M, τ) has property (T) if and only if thereexists ε > 0 and a finite subset F ⊂M such that every M -M -bimodule Hthat has a unit vector ξ satisfying ‖xξ−ξx‖ ≤ ε, for all x ∈ F , actually hasa non-zero M -central vector ξ0, meaning that xξ0 = ξ0x, for all x ∈M .

One can prove that an ICC group Γ has property (T) if and only if theII1 factor L(Γ) has property (T) in the sense of Connes and Jones; seefor example [55, Proposition B.5]. This results also holds when L(Γ) istwisted by a scalar 2-cocycle. This result illustrates the existence of adictionary between the category of countable groups and the category ofII1 factors: L(Γ)-L(Γ)-bimodules over the II1 factor L(Γ) are the analogueof unitary representations of the countable group Γ. More precisely, givena unitary representation π : Γ → U(Hπ) we can build the following L(Γ)-L(Γ)-bimodule H defined by

H := `2(Γ)⊗Hπ ,


where

ug · (δh ⊗ ξ) · uk := δghk ⊗ π(g)ξ, for all g, h, k ∈ Γ, ξ ∈ Hπ .

Such dictionary and further links between countable groups, actions ofcountable groups on standard probability spaces and type II1 factors arevery clearly explained in Connes’ Bourbaki seminar [4].

Definition 2.11. Let (M, τ) be a von Neumann algebra endowed with afaithful normal tracial state τ . Let H be an M -M bimodule.

1. A sequence of vectors (ξn) ∈ H is said to be almost central if

‖xξn − ξnx‖ → 0, for all x ∈M .

2. A sequence of vectors (ξn) ∈ H is said to be almost tracial if

|〈ξn, ξnx〉 − τ(x)| → 0, and |〈ξn, xξn〉 − τ(x)| → 0, for all x ∈M .

In [36], Popa defines a notion of relative Property (T) for inclusions B ⊂M of tracial von Neumann algebras which is stated as follows.

Definition 2.12. Let (M, τ) be a von Neumann algebra endowed witha faithful normal tracial state τ and B ⊂M a von Neumann subalgebra.The pair (B ⊂M) is said to have the relative Property (T) (or to be rigid)if the following holds. Every M -M -bimodule that admits a sequence ofalmost-central, almost tracial unit vectors, admits a sequence of almosttracial and B-central unit vectors.

Remark 2.13. We have the trivial observation that if B is a property(T) II1 factor, every inclusion B ⊂M , where M is a tracial von Neumannalgebra is rigid.

Note for later use that the condition of almost-traciality and almost cen-trality can be replaced by a “mixed condition”, as follows.

Lemma 2.14. Let (M, τ) be a von Neumann algebra with a faithful nor-mal tracial state τ . Let H be an M -M -bimodule. Let (ξn) ∈ H be asequence of unit vectors. The following are equivalent.

• The sequence (ξn) is almost central and almost tracial.

2.4 Outer actions of countable groups on II1 factors 41

• |〈ξn, xξny〉 − τ(xy)| → 0, for all x, y ∈M .

Proof. Let x, y ∈ M . Suppose that (ξn)n is almost central and almosttracial.

|〈ξn, xξny〉 − τ(xy)| = |〈ξny∗, xξn〉 − τ(xy)|6 |〈y∗ξn, xξn〉 − τ(xy)|+ |〈ξny∗ − y∗ξn, xξn〉|6 |〈ξn, yxξn〉 − τ(xy)|+ ‖x‖‖ξny∗ − y∗ξn‖ .

which goes to 0 by definition of (ξn).

Suppose that the sequence (ξn) satisfies the second assertion of the lemma.We prove that (ξn) is almost tracial. Let x ∈M . Then,

‖xξn − ξnx‖2 = 〈ξn, x∗xξn〉+ 〈ξn, ξnxx∗〉 − 2<(〈ξn, x∗ξnx〉

)=(〈ξn, x∗xξn〉 − τ(x∗x)

)+(〈ξn, ξnxx∗〉 − τ(xx∗)

)− 2<

(〈ξn, x∗ξnx〉 − τ(x∗x)

),

which goes to zero, by the second assertion.

2.4 Outer actions of countable groups on II1 fac-

tors

In this section, we gather some well known results concerning outer ac-tions of countable groups on type II1 factors that we will use in thesequel. In this section Γ always denotes a countable group and N a typeII1 factor.

Definition 2.15. Let Γ be a countable group and N a II1 factor. Wesay that a homomorphism σ : Γ → Aut(N) defines an outer action of Γon N if σg is not an inner automorphism of N , for all g 6= e.

Every countable group admits an outer action on the hyperfinite II1 fac-tor, via non-commutative Bernoulli shifts Γ y

⊗g∈ΓR.

The following lemma gives a useful criterion to prove that an action isouter and proves that in the case of factors, outer actions are the properlyouter actions in the sense of definition 1.11.


Lemma 2.16. Let Γ be a countable group and N a factor. Then, σ :Γ y N is outer if and only if for all x ∈ N and all g 6= e,(

xy = σg(y)x, for all y ∈ N)

=⇒ x = 0 .

Proof. Suppose that the equality xy = σg(y)x holds for all y ∈ N , forsome non-zero element x ∈ N and g 6= e. Then, xx∗ and x∗x are elementsof N that commute with N . So, by factoriality, there exists a non-zeroλ ∈ R+ such that xx∗ = x∗x = λ1. Note that we could choose the samescalar λ for xx∗ and x∗x since both of these elements have the same norm.Dividing x by

√λ yields a unitary u such that σg = Adu, contradicting

the outerness of the action Γ on N .

The other implication is easy because if there exists a non-trivial elementg ∈ Γ such that σg is inner, then there exists a unitary x ∈ N such thatσg(y)x = xy, for all y ∈ N . Since x is a unitary, x is non-zero.

Let Γ y N outerly. The resulting crossed-product N o Γ is a factorand more precisely, the inclusion N ⊂ N o Γ is irreducible meaning thatN ′ ∩N o Γ = C1.

Lemma 2.17. Let σ : Γ y N be an outer action. Then, the normalizerof N ⊂ N o Γ can be described as follows.

NormNoΓ(N) = xug | x ∈ U(N), g ∈ Γ .

Proof. Let M := N o Γ. Every element w ∈ NormM (N) yields an auto-morphism β ∈ Aut(N) such that

wx = β(x)w, for all x ∈ N . (2.2)

Then, writing w as the L2-convergent sum w =∑

g∈Γwgug, formula (2.2)gives

wgσg(x) = β(x)wg, for all g ∈ Γ, x ∈ N . (2.3)

Let g ∈ Γ such that wg 6= 0. Equation (2.3) and the factoriality of Nimply wg is a multiple of a unitary in N so we may assume that wg is aunitary. Suppose that wg and wh are two such unitaries. Then, formula(2.3) yields

w∗gwhσh(x) = σg(x)w∗gwh, for all g ∈ Γ, x ∈ N .

So w∗gwh = σgh−1(x)w∗gwh, for all x ∈ N . By outerness of the action of Γon N , we have g = h, which ends the proof.

2.4 Outer actions of countable groups on II1 factors 43

Notation 2.18. Let M be a von Neumann algebra and N ⊂ M a vonNeumann subalgebra. We denote by Aut(N ⊂ M) the subgroup ofAut(M) consisting of automorphisms of M leaving N globally invariant.

Proposition 2.19. Let σ : Γ y N be an outer action and α ∈ Aut(N ⊂N o Γ). Then, there exists an automorphism δ ∈ Aut(Γ) and a mapx : Γ → U(N) such that

• xgh = xgσg(xh), for all g, h ∈ Γ and

• α(ug) = xδ(g)uδ(g), for all g ∈ Γ.

Proof. Since α is an automorphism of N o Γ such that α(N) = N theunitaries α(ug) normalize N inside N o Γ. So, because of Lemma 2.17,there exists a unique element δ(g) ∈ Γ and a unitary xδ(g) in N suchthat α(ug) = xδ(g)uδ(g), for all g ∈ Γ. The uniqueness of the element δ(g)comes from the proof of Lemma 2.17, so in particular, the map δ : Γ → Γis bijective. We compute, for all g, h ∈ Γ

α(uguh) = α(ug)α(uh)

= xδ(g)uδ(g)xδ(h)uδ(h)

= xδ(g)σδ(g)(xδ(h))uδ(g)δ(h) .

On the other hand, α(ugh) = xδ(gh)uδ(gh), so we are done.

Remark 2.20. Let G be a locally compact group acting on the vonNeumann algebra M . A continuous map x : G → U(M) is called a1-cocycle (with values in M) if it satisfies the relation

xgh = xgσg(xh) for all g, h ∈ G .

A 1-cocycle is called a coboundary if there exists a unitary x ∈ M suchthat

xg = x∗σg(x), for all g ∈ G .

For example, if G y M is a minimal action of the compact group G onthe II1 factor M , then all 1-cocycles with values in M are coboundaries(see [60]). See Chapter 7 concerning minimal actions.

Proposition 2.21. Let Γ y N outerly and α ∈ Aut(N o Γ) such thatα(x) = x, for all x ∈ N . Then, there exists a character ω ∈ Char Γ suchthat

α(ug) = ω(g)ug, for all g ∈ Γ .


Proof. Let g ∈ Γ and x ∈ N . Then,

α(ug)u∗gx = α(ug)σg−1(x)u∗g= α(ugσg−1(x))u∗g= α(xug)u∗g= xα(ug)u∗g .

Then, α(ug)u∗g ∈ N o Γ∩N ′ = C1, by outerness of the action of Γ on N .So there exists a map ω : Γ → S1 such that α(ug) = ω(g)ug. It is easilychecked that ω is a character of Γ.

Notation 2.22. We use the following notation throughout the text.Given an outer action Γ y N and a character ω of the group Γ, wealways denote by αω the automorphism αω ∈ Aut(N o Γ) such thatαω(ug) = ω(g)ug and αω(x) = x, for all x ∈ N .

2.5 The ∗-algebra of operators affiliated with a

II1 factor

We recall results and notations concerning affiliated operators that willbe often used in the sequel. In this section M ⊂ B(H) is a II1 factor withnormal tracial state τ .

We refer to [50] for the necessary background concerning unbounded oper-ators on Hilbert space and their connection with von Neumann algebras.We use the following notations. Given two unbounded operators S andT on a Hilbert space H, with domains dom(S) and dom(T ), we say thatT is an extension of S and write S ⊂ T whenever dom(S) ⊂ dom(T ) andthe operator T satisfies T| dom(S) = S.

Definition 2.23. Let (M, τ) ⊂ B(H) be a II1 factor

• An operator T : dom(T ) → H is said to be affiliated with M if itsatisfies xT ⊂ Tx, for every x ∈M ′ ∩ B(H).

• A densely defined closed operator T affiliated with M , with dom(T )⊂ L2(M, τ), is said to be square-integrable if 1 ∈ dom(T ).

2.5 The ∗-algebra of operators affiliated with a II1 factor 45

Proposition 2.24. Let (M, τ) be a II1 factor. Define, for every ξ ∈L2(M), the densely defined operator affiliated with M given by

L0ξ : L2(M) → L2(M) : a 7→ ξ · a .

Then, L0ξ is closable and its closure Lξ is a square integrable operator.

Proof. We prove that the operator L0ξ is closable so equivalently, that

(L0ξ)∗ is densely defined. Recall the following formulas, coming from the

GNS construction.

J(ξx) = J(x∗ξ), and (Jx∗J)ξ = ξx, for all ξ ∈ L2(M), x ∈M . (2.4)

We prove now that L0Jξ ⊂ (L0

ξ)∗. For every x, y ∈M , we have that

〈L0ξx, y〉 = 〈ξx, y〉

= 〈y∗ξ, x∗〉= 〈Jx∗, Jy∗ξ〉= 〈x, J(ξ)y〉 = 〈x, T 0

J(ξ)y〉 .

The operator (L0ξ)∗ is hence densely defined. Identifying M ′ ∩B(L2(M))

with JMJ we obtain, for all y ∈M and x ∈ dom(L0ξ),

Jy∗JL0ξ(x) = ξ · xy

= L0ξ(Jy

∗J)x ,

proving that L0ξ is affiliated withM . Define Lξ to be the closure of L0

ξ . Letx ∈ dom(Lξ). By definition of Lξ, we can take a sequence (xn) ∈ dom(L0

ξ)converging to x such that L0

ξ(xn) converges to Lξ(x). Since L0ξ is affiliated

to M , the operators Lξ follows affiliated to M , by taking limits.

One can obtain a stronger result by proving that the map ξ 7→ Lξ gives anone to one correspondence with L2(M) and the space of square integrableoperators affiliated with M ; see [3, Proposition F.11] for a complete proof.Furthermore, denote by M the closed densely defined operators affiliatedwith M . By [30, Theorem XV, page 229], we know that M is a ∗-algebra,where sum and product are defined as the closure of sum and producton the natural domains and where the adjoint is the usual adjoint ofoperators. Denote by M+ the positive self-adjoint operators affiliated


with M . Then, τ has a natural extension to a positive-linear map M+ →[0,+∞]. Define, for x ∈ M, |x| := (x∗x)1/2, ‖x‖2 := τ(x∗x)1/2 and‖x‖1 := τ(|x|). Then, one can prove that

L2(M) = x ∈M | ‖x‖2 <∞ .

Define similarly

L1(M) := x ∈M | ‖x‖1 <∞ .

Actually, L1(M) is the linear span of x ∈ M+ | τ(x) < ∞ and τ

extends to a linear map L1(M) → C. Both L2(M) and L1(M) are stableunder the adjoint and are M -M -bimodules. Finally, the product of twoelements in L2(M) belongs to L1(M) and the Cauchy-Schwarz inequalityholds.

Every x ∈ M has a unique polar decomposition, x = u|x|, where u is apartial isometry in M with u∗u equal to the support projection of |x|. IfN ⊂ M is an irreducible subfactor, meaning that N ′ ∩M = C1, everyelement x ∈ M satisfying ax = xa for all a ∈ N , belongs to C1 as well.Indeed, if x = u|x| is the polar decomposition of x, the uniqueness of thepolar decomposition implies that u and |x| commute with all unitaries inN . Hence, u is scalar. By the uniqueness of the spectral decompositionof |x|, all spectral projections of |x| are scalar and hence, also |x| followsscalar.

Part II

The category of bimodules

over a type II1 factor

Chapter 3

An introduction to

bimodules

This chapter consists of a review of classical facts concerning bimodulesover II1 factors. The material gathered here is now well known and onlyincluded in this text for the reader’s convenience. We claim no originalityin our proofs that are essentially taken from [1]. In [1], connections withbimodules and subfactor theory are also investigated. For further readingon bimodules, also called Connes correspondences, we refer to the bookof Connes [5] and Popa’s notes [40]. Throughout this section, N , M andP denote type II1 factors, unless explicitly mentioned otherwise.

The definition of bimodules has been given in Definition 2.8. We introducenow the space Mor(H,K) of intertwiners between bimodules H and K.This space, which turns to be a von Neumann algebra when H = K, is avery important space for the analysis of bimodules, we will come back toit more closely when discussing irreducible decomposition and Frobeniusreciprocity.

Definition 3.1. Let H and K be M -N -bimodules. The space

MorM (H,K)N := T ∈ B(H,K) | T (a · ξ · b) = a · T (ξ) · b,a ∈M, b ∈ N, ξ ∈ H

is called the space of M -N -intertwiners or also M -N -bimodular maps.When H = K, we simply write MorM (H)N .

50 Chapter 3. An introduction to bimodules

We sometimes only consider the right or left intertwiners space that we de-note respectively by Mor(H,K)N and MorM (H,K). We will say that twoM -N -bimodules H and K are equivalent when MorM (H,K)N contains aunitary. We will always identify two bimodules that are equivalent. Bi-jective elements in MorM (H)N are called M -N -bimodule isomorphisms.In fact, it is only necessary to consider M -N -bimodule isomorphismsT : H → K to prove that H and K are actually equivalent. Indeed, thepolar part of T yields the required unitary in MorM (H,K)N .

3.1 Direct sums of modules.

Let (Hi)i∈I a family of M -N -bimodules. The Hilbert space K :=⊕

i∈I Hi

is turned into a bimodule with the following actions

a ·

(⊕i∈I

ξi

)· b :=

⊕i∈I

a · ξi · b, for all a ∈M, b ∈ N, (ξi) ∈ K .

The notion of direct sum of bimodules leads us to the notion of subbi-module and irreducible bimodule.

Definition 3.2. Let H be an M -N -bimodule.

• A closed subspace K ⊂ H is an M -N -subbimodule of K if MKN ⊂K.

• The bimodule H is said to be irreducible when H has no non-trivialproper subbimodule.

The irreducibility ofH can be expressed in terms of the space of intertwin-ers. Indeed if H admits a non-trivial orthogonal decomposition then theprojections on each summand yield non-trivial elements in MorM (H)N .On the other hand, the image of a non trivial projection in MorM (H)Nand its orthogonal complement in H yield a non-trivial decomposition ofH so we obtain the following lemma.

Lemma 3.3. Let H be an M -N -bimodule. Then, H is irreducible if andonly if MorM (H)N = C1.

3.2 Connes’ tensor product of bimodules 51

3.2 Connes’ tensor product of bimodules

We present the construction and some basic properties of Connes’ tensorproduct of bimodules. More information can be found in Connes’ book[5, V.Appendix B].

Definition 3.4. Let H be a right N -module. For every ξ ∈ H we havethe densely defined operator given by

Lξ : L2(N) → H : a 7→ ξ · a, for all a ∈ N .

The elements of the subset

H0 := ξ ∈ H | Lξ extends to a bounded operator L2(N) → H

are called right-bounded vectors.

Lemma 3.5. Let H := L2(N). Then, H0 = N .

Proof. The operator Lξ commutes to the right module action and hencedefines an element of N . Since, for every x ∈ N, Lx = x and Lξ(1) = ξ,the map ξ 7→ Lξ is a well defined isomorphism between H0 and N .

The set of bounded vectors H0 of a right module H is always dense in H.

We define an N -valued inner product on the set of bounded vectors bythe formula

〈ξ, η〉N := L∗ξLη .

It is easily checked that L∗ξLη commutes with the right action of N onL2(N) and thus defines an element of N .

Lemma 3.6. The N -valued scalar product 〈·, ·〉N satisfies the followingproperties.

1. (ξ, η) 7→ 〈ξ, η〉N is a positive sesquilinear map,

2. (〈ξ, η〉N )∗ = 〈η, ξ〉N ,

3. 〈ξ · a, η〉N = a∗〈ξ, η〉N ,

4. 〈ξ, η · a〉N = 〈ξ, η〉N a.


Note that item 4 in the foregoing lemma is a consequence of 2 and 3.The following lemma shows that the N -valued inner product 〈ξ, η〉N canbe seen as the Radon-Nikodym derivative of a 7→ 〈ξ, η · a〉. The proof isstraightforward.

Lemma 3.7. Let H be right N -module on the II1 factor (N, τ). Then,

〈ξ, η · a〉 = τ(〈ξ, η〉Na), for all a ∈ N, ξ, η ∈ H0 . (3.1)

As a consequence, we immediately obtain that

τ(〈x, y〉N ) = 〈x, y〉 .

Lemma 3.8. Let H be a right N -module and K a left Nbimodule. Theformula

〈a⊗ ξ, b⊗ η〉 := 〈ξ, 〈a, b〉N · η〉 (3.2)

defines a positive sesquilinear form on H0 K.

Proof. The only non-trivial point to be checked is the positivity of thisform. We have⟨

n∑i=1

ξi ⊗ ηi,n∑j=1

ξj ⊗ ηj

⟩=

∑1≤i,j≤n

〈ηi, 〈ξi, ξj〉N · ηj〉 .

We only need to prove that the operator T = (〈ξi, ξj〉N )1≤i,j≤n is positive,for every ξ1, . . . , ξn ∈ H0. Indeed, if it is the case and if η := (η1, . . . , ηn)ᵀ,

0 ≤ 〈η, Tη〉 =∑

1≤i,j≤n〈ηi, 〈ξi, ξj〉N · ηj〉 .

Let Ξ := (ξ1, . . . , ξn). We have LΞ : L2(Nn) → (Cn)∗ ⊗H. Note that theGNS inner product of two elements a = (aij) ∈ Nn and b = (bjk) ∈ Nn

is given by

(Tr⊗τ)(a∗b) =∑i,j

τ(a∗jibij) =∑i,j

〈aji, bji〉 .


We compute the adjoint of LΞ. Let η1, . . . , ηn ∈ H0.

〈(ξ1, . . . , ξn) · a, (η1, . . . , ηn)〉

=

⟨(∑i

ξi · ai1, . . . ,∑i

ξi · ain

), (η1, . . . , ηn)

⟩=∑i,j

〈ξi · aij , ηj〉

=∑i,j

〈aij , L∗ξiηj〉 .

Then, L∗Ξ(η1, . . . , ηn) = (L∗ξiηj)i,j . So, T = L∗ΞLΞ = 〈Ξ,Ξ〉Nn , which ispositive, by Lemma 3.6.

We have now all ingredients to define Connes’ tensor product.

Definition 3.9. Let H be a right N -bimodule and K a left N -bimodule.Connes tensor product is defined as the separation and completion of thealgebraic tensor product H0K for the sesquilinear form defined in (3.2)and is denoted H⊗N K.

The precise construction goes as follows. Let N〈·,·〉 denote the null spaceof the sesquilinear form defined in (3.2) and ρr : H0 K → H0K

N〈·,·〉, the

quotient map. The quotient space is a pre-Hilbert space, endowed withthe inner product

〈ρr(ξ ⊗ η), ρr(ξ′ ⊗ η′)〉 := 〈ξ ⊗ η, ξ′ ⊗ η′〉 , (3.3)

which is well defined, by Cauchy-Schwarz inequality. Then, H ⊗N K isthe completion of H0K

N〈·,·〉for the norm induced by this inner product.

Notation 3.10. We will write ξ ⊗N η := ρr(ξ ⊗ η).

We have defined the notion of right-bounded vectors of a right module. IfK is a left N -module, we have, analogously, the densely defined operators

Rξ : L2(N) → K : a 7→ a · ξ

and the the set 0K of left-bounded vectors for the left module K definedby

0K := ξ ∈ K | Rξ extends to a bounded operator L2(N) → K .


Denote by J the canonical anti-unitary operator on L2(M) and define

N〈ξ, η〉 := J(R∗ξRη)∗J, for all ξ, η ∈ 0K .

It is easily checked that (ξ, η) 7→ 〈ξ, η〉N is a positive, conjugate sesquilin-ear map that commutes to the right action of N on L2(N) and thus de-fines an element of N . The left version of Lemma 3.6 yields the followingproperties.

Lemma 3.11. 1. (N〈ξ, η〉)∗ = N〈η, ξ〉,

2. N〈a · ξ, η〉 = aN〈ξ, η〉,

3. N〈ξ, a · η〉 = N〈ξ, η〉a∗.

We also have the left Radon-Nikodym derivative formula

〈ξ, a · η〉 = τ(aN〈ξ, η〉

). (3.4)

The set of left bounded vectors 0K is dense in K.

Example 3.12. Let H := L2(N). Then, H0 = 0H = N , 〈a, b〉N = a∗b

and N〈a, b〉 = ba∗.

Let H be a right N -module and K be a left N -module. As in Lemma3.8, we define a positive sesquilinear form on the algebraic tensor productH 0K given by

〈ξ ⊗ a, η ⊗ b〉 = 〈ξ, η · N〈a, b〉〉 . (3.5)

In the following proposition, we prove that the Connes tensor productH⊗N K can be unambiguously defined as the separation and completionof either H0K, for the sesquilinear form defined in (3.2), either H 0K,for the sesquilinear for defined in (3.5).

Proposition 3.13. Let H be a right N -module and K a left N -module.

• The sesquilinear forms defined in (3.2) and (3.5) both agree on H00K.

• Separation and completion of H0 K, H 0K and H0 0K are allequal to H⊗N K.


Proof. Let ξi, ξ′i ∈ H0 and ηi, η′i ∈ 0K, for i = 1, . . . , n. Denote by Φr andΦl the sesquilinear forms defined in (3.2) and (3.5). Then,

Φr

∑i

ξi ⊗ ηi,∑j

ξ′j ⊗ η′j

=∑i,j

⟨ηi, 〈ξi, ξ′i〉N · η′j

⟩=

∑i,j

τ(〈ξi, ξ′i〉NN〈ηi, η′j〉), by (3.4)

=∑i,j

〈ξi, ξ′i · N〈ηi, η′j〉〉, by (3.1)

= Φl

∑i

ξi ⊗ ηi,∑j

ξ′j ⊗ η′j

.

Since H0 is dense in H and 0K is dense in K, we easily prove that

H0 0Kφr = H0 Kφr and H0 0Kφl = H 0Kφl.

Since φl and φr coincide on H0 0K, we have proven that all threeconstructions give rise to the same tensor product H⊗N K.

Let H be an M -N -bimodule and K an N -P -bimodule. There is a naturalstructure of M -P -bimodule on H⊗N K that we define now.

Proposition 3.14. Let H be an M -N -bimodule and K an N -P -bimodule.Then, the Hilbert space H⊗NK is an M -P -bimodule for the left M -actionand right P -action given by

a · (b⊗ ξ) = ab⊗ ξ and (b⊗ ξ) · a = b⊗ (ξa) .

Proof. Let ξ1, . . . , ξn ∈ H0, η1, . . . , ηn ∈ K and a ∈ P . We define

Ξ := (ξ1, . . . , ξn) and η := (η1, . . . , ηn)ᵀ .

We also define the operator T := (〈ξi, ξj〉N )i,j ∈ Mn(C) ⊗ N . We haveseen that T = L∗ΞLΞ, in the proof of Lemma 3.8. In particular, T is


positive. Then, we obtain that∥∥∥∥∥∑i

ξi ⊗ ηia

∥∥∥∥∥2

=∑i,j

〈ηia, 〈ξi, ξj〉N · (ηja)〉

= 〈η · a, Tη · a〉

= 〈T 1/2η, T 1/2η · aa∗〉

≤ ‖a‖2〈T 1/2η, T 1/2η〉

= ‖a‖2

∥∥∥∥∥∑i

ξi ⊗ ηi

∥∥∥∥∥2

.

Using Proposition 3.13, we can now easily prove that Connes’ tensorproduct is associative.

Lemma 3.15. Let H be an M -N -bimodule, K an N -P -bimodule and La P -Q-bimodule. Then, the obvious map

MH⊗N (K ⊗P LQ) → (MH⊗N K)⊗P LQ

is an M -Q-bimodular unitary.

Proof. We define

u : H0 (K 0L) → (H0 K) 0L ,

such that u(ξ ⊗ (η ⊗ µ)

)= (ξ ⊗ η)⊗ µ. Then, using the associativity of

the algebraic tensor product, we obtain that

〈(ξ ⊗ η)⊗ µ, (ξ′ ⊗ η′)⊗ µ′〉 =⟨η, 〈ξ, ξ′〉N · η′ · P〈µ, µ′〉

⟩= 〈ξ ⊗ (η ⊗ µ), ξ′ ⊗ (η′ ⊗ µ′)〉 .

So u is an isometry and because of Proposition 3.13 it extends to a unitaryu : MH⊗N (K ⊗P LQ) → (MH⊗N K)⊗P LQ.

Connes tensor product is also a distributive operation on direct sums ofbimodules.


Lemma 3.16. Let H,K be M -N -bimodules and L an N -P -bimodule.Then, the obvious map(

MHN ⊕MKN)⊗N LP →

(MH⊗N LP

)⊕(MK ⊗N LP

)is an M -P -bimodular unitary.

Proof. We define

u : (H⊕K)0 L0 → (H0 L0)⊕ (K0 L0) ,

such that u((ξ⊕ η)⊗ µ) = (ξ⊗ µ)⊕ (η⊗ µ). An immediate computationproves that (H⊕K)0 = H0 ⊕K0 and

〈ξ ⊕ η, ξ′ ⊕ η′〉N = 〈ξ, ξ′〉N + 〈η, η′〉N ,

for every ξ, ξ′ ∈ H0 and η, η′ ∈ K0. Then, we have, for every ξ, ξ′ ∈ H0,η, η′ ∈ K0 and µ, µ′ ∈ L0,

〈(ξ ⊗ µ)⊕ (η ⊗ µ), (ξ′ ⊗ µ′)⊕ (η′ ⊗ µ′)〉= 〈ξ ⊗ µ, ξ′ ⊗ µ′〉+ 〈η ⊗ µ, η′ ⊗ µ′〉= 〈µ, 〈ξ, ξ′〉N · µ′〉+ 〈µ, 〈η, η′〉N · µ′〉= 〈µ,

(〈ξ, ξ′〉N + 〈η, η′〉N

)· µ′〉

= 〈µ, 〈ξ ⊕ η, ξ′ ⊕ η′〉N · µ′〉= 〈(ξ ⊕ η)⊗ µ, (ξ′ ⊕ η′)⊗ µ′〉 .

We obtain a unitary, by the usual arguments.

Lemma 3.17. Let MHN be an M -N -bimodule. We have the followingisomorphisms of M -N -bimodules:

MH⊗N L2(N)N ∼= MHN ∼= ML2(M)⊗M HN .

Moreover, if M ⊂ N then,

MH⊗N L2(N)M ∼= MHM .

Proof. We only prove the first equivalence of M -N -bimodules, the othersfollow similarly. Since we have 〈a, b〉M = a∗b, for all a, b ∈ M . Thefollowing computations prove that the map

T : M H → H : (a⊗ ξ) 7→ aξ


is a surjective isometry that factors in an isomorphism ML2(M)⊗MHN∼=

MHN. For every a1, . . . , an ∈M and ξ1, . . . , ξn ∈ H we have that∑i,j

〈aiξi, ajξj〉 =∑i,j

〈ξi, 〈ai, aj〉N · ξj〉

=∥∥∑

i

ai ⊗ ξi∥∥2.

3.3 Contragredient bimodule

If MHN is an M -N -bimodule, the contragredient bimodule NHM is de-fined on the conjugate Hilbert space H = H∗ with bimodule actions givenby

a · ξ = ξa∗ and ξ · a = a∗ξ .

Lemma 3.18. Let MHN be an M -N -bimodule. Then, a vector ξ belongsto H0 if and only if ξ ∈ 0H and

N〈ξ, η〉 = 〈η, ξ〉N .

Proof. For all a, b ∈ N , we have, using the Radon Nikodym formulas (3.1)and (3.4)

τ(a N〈ξ, η〉) = 〈ξ, a · η〉= 〈ξ, η · a∗〉= 〈η · a∗, ξ〉= 〈η, ξ · a〉= τ(a〈η, ξ〉N ) .

Example 3.19. Let N ⊂ M be an inclusion of II1 factors. The canon-ical anti-unitary map J on L2(M) implements the isomorphism of N -M -bimodules M L2(M)N ∼= N L2(M)M . Note that in particular theM -M -bimodule L2(M) is isomorphic to its contragredient. Bimoduleswith such property are called self-contragredient.

3.4 Intertwiners 59

Proposition 3.20. Let H be an M -N -bimodule and K an N -P -bimodule.The map

T : K0 ⊗N 0H → H⊗N K : η ⊗ ξ 7→ ξ ⊗ η

extends to a P -M -bimodule isomorphism K ⊗N H ∼= H⊗N K.

Proof. Let ξ, ξ′ ∈ H0 and η, η′ ∈ 0K. Then⟨ξ ⊗ η, ξ′ ⊗ η′

⟩=

⟨ξ′ ⊗ η′, ξ ⊗ η

⟩= 〈ξ′, ξ · N〈η′, η〉〉= 〈ξ′, ξ · 〈η, η′〉N 〉= 〈ξ′ · (〈η, η′〉N )∗, ξ〉

= 〈ξ, ξ′ · (〈η, η′〉N )∗〉= 〈ξ, 〈η, η′〉N · ξ′〉=

⟨η ⊗ ξ, η′ ⊗ ξ′

⟩.

The above computations prove that we have defined a surjective isometrythat factors in a unitary u : K ⊗N H → H⊗N K. This unitary is alsoP -M -bimodular since for all a ∈ P and b ∈M

u(a · (η ⊗ ξ) · b) = u(η · a∗ ⊗ b∗ · ξ)= b∗ · (ξ ⊗ η) · a∗

= a · (ξ ⊗ η) · b .

3.4 Intertwiners

Let H,K be Hilbert spaces. To every bounded linear map T : H → K

we associate the bounded linear map T : H → K defined by

T (ξ) = T (ξ) .

Note that if L is another Hilbert space and S : K → L a bounded linearmap, we have that S T = S T .

Proposition 3.21. Let H,K be M -N -bimodules. Then, the map T 7→ T ∗

is an isomorphism

MorM (H,K)N ∼= MorN (K,H)M .


Proof. Let T ∈ MorM (H,K)N . Then T ∗ ∈ MorN (K,H)M . Indeed,

T ∗(a · η · b) = T ∗(b∗ · η · a∗)= T ∗(b∗ · η · a∗)= b∗ · T ∗(η) · a∗

= a · T ∗(η) · b = a · T ∗(η) · b .

Similarly, every T ∈ MorN (K,H)M yields an element T ∗ ∈ MorM (H,K)Nso the map T 7→ T ∗ is the required isomorphism.

Lemma 3.22. Let H and K be M -N -bimodules and T ∈ MorM (H,K)N .Then, TH0 ⊂ K0 and we have

〈T (ξ), T (η)〉N = 〈ξ, T ∗T (η)〉N = 〈T ∗T (ξ), η〉N .

Proof. Let ξ be a bounded vector in H. Then, there exists a constantC > 0 such that ‖ξ · a‖ ≤ C‖a‖2. So, ‖T (ξ) · a‖ ≤ C‖T‖‖a‖2 and T (ξ) isbounded in K.

Let a, b ∈ N and ξ, η ∈ H0. Then,⟨〈T (ξ), T (η)〉Na, b

⟩=⟨LT (η)(a), LT (ξ)(b)

⟩= 〈T (η) · a, T (ξ) · b〉= 〈η · a, T ∗T (ξ) · b〉= 〈L∗T ∗T (ξ)Lη(a), b〉 .

The other equality can be proven in the same way.

Proposition 3.23. Let H1,K1 be M -N -bimodules and H2,K2 be N -P -bimodules. Let S ∈ MorM (H1,K1)N and T ∈ MorN (H2,K2)P . Thereexists a unique intertwiner, denoted S ⊗N T ∈ MorM (H1 ⊗N H2,K1 ⊗NK2)P such that

(S ⊗N T )(ξ ⊗ η) = S(ξ)⊗ T (η), for all ξ ∈ H01, η ∈ H0

2 .

Proof. Note that the map S⊗N T defined in the proposition is necessarilyunique, since H0

1 H02 is dense in H1 ⊗N H2. We construct the elements

S ⊗N 1 and 1⊗N T defined by

(S ⊗N 1)(ξ ⊗ η) := S(ξ)⊗ η, for every ξ ∈ H01, η ∈ H2 ,

3.4 Intertwiners 61

and

(1⊗N T )(ξ ⊗ η) := ξ ⊗ T (η), for every ξ ∈ H1, η ∈ H02 .

Then, S ⊗N T := (S ⊗N 1)(1 ⊗N T ) will be the required element ofMorM (H1 ⊗N H2,K1 ⊗N K2)P .

Construction of S ⊗N 1. Let ξ1, . . . , ξn ∈ H01 and η1, . . . , ηn ∈ H2. Define

Ξ := (ξ1, . . . , ξn) and η := (η1, . . . , ηn)ᵀ .

We start by proving the following inequality in Mn(C)⊗N .

〈SΞ, SΞ〉Nn ≤ ‖S‖2〈Ξ,Ξ〉Nn . (3.6)

Since S is right N -linear, we have that LSΞ = SLΞ. Then, L∗SΞLSΞ =L∗ΞS

∗SLΞ. So, we only have to prove that L∗ΞS∗SLΞ ≤ ‖S‖2〈Ξ,Ξ〉Nn .

Let a ∈ Nn. Then, we obtain

〈L∗ΞS∗SLΞ(a), a〉 = 〈S∗SLΞ(a), LΞ(a)〉≤ ‖S‖2〈LΞ(a), LΞ(a)〉=

⟨‖S‖2〈Ξ,Ξ〉Nn · a, a

⟩,

and formula (3.6) is proven. We prove now that S ⊗ 1 factors in a con-tinuous map S ⊗N 1 : H1 ⊗N H2 → K1 ⊗N K2.∥∥∥∥∥(S ⊗ 1)

(∑i

ξi ⊗ ηi

)∥∥∥∥∥2

=∑i,j

⟨ηi, 〈S(ξi), S(ξj)〉N · ηj

⟩= 〈η, 〈SΞ, SΞ〉Nn · η〉≤ ‖S‖2〈η, 〈Ξ,Ξ〉Nn · η〉= ‖S‖2

∑i,j

⟨ηi, 〈ξi, ξj〉N · ηj

⟩= ‖S‖2

∥∥∥∥∥∑i

ξi ⊗ ηi

∥∥∥∥∥2

.

Similar computations on H1 H2 prove that we have a continuous map1⊗N T : H1 ⊗N H2 → K1 ⊗N K2.

Chapter 4

Coupling constant, Jones

index

In this chapter we recall some classical facts concerning the dimension ofmodules over II1 factors and the Jones index. Our goal is not to providea detailed treatment of these subjects but to include, for the reader’sconvenience, some important results and formulas that will be used lateron. Most of the proofs are not written and many results are omitted, werefer to [25] for further details and links with subfactor theory.

Here M always denotes a type II1 factor, endowed with its trace τM .When there is no ambiguity, we simply write τ for the trace on M .

4.1 Dimension of modules over a II1 factor

We fix some notations and recall some results from the GNS construction,see section 1.4. We denote by λ the left representation of M on L2(M)and by ρ the right anti-representation of M on L2(M). Then, we havethat

ρ(M)′ ∩ B(L2(M)) = λ(M) . (4.1)

We know that a right or left M -module H is the data of a normal repre-sentation of Mop or M on B(H). Classifying right modules then amountsto classifying representations of M . When M is a II1 factor (rememberthat we assumed that all von Neumann algebras have separable predual

64 Chapter 4. Coupling constant, Jones index

in this thesis), representations of M are classified by a number between0 and +∞. The key result to understand this classification is the follow-ing proposition, see [22, Chapter 10]. Every representation of M can becompared to the amplification 1⊗ λ of λ on `2(N)⊗ L2(M).

Proposition 4.1. Let M be a II1 factor and π : M → B(H) a represen-tation. There exists an isometry u : H → `2(N)⊗ L2(M) such that

uπ(x) = (1⊗ λ(x))u, for all x ∈M .

As as immediate consequence, we obtain

Proposition 4.2. Let H be a right M -module. Then, H is a projectivemodule: there exists a projection p ∈ B(`2(N))⊗M such that

HM∼=(p · (`2(N)⊗ L2(M))

)M .

Proof. Denote by πr : M → B(H) the normal anti-representation definingthe right M -action on H. Then, Proposition 4.1, applied to the oppo-site algebra Mop, yields an isometry u : H → `2(N) ⊗ L2(M) such thatuπr(x) = (1 ⊗ ρ(x))u, for all x ∈ M . Then, uu∗ ∈ B

(`2(N)

)⊗(ρ(M)′ ∩

B(L2(M)). So there exists a projection p ∈ B(`2(N))⊗M such that

uu∗ = (id⊗ λ)(p) and

H ∼= (id⊗ λ)(p)(`2(N)⊗ L2(M)

)= p ·

(`2(N)⊗ L2(M)

),

as right M -modules.

We can now define the dimension of a right M -module.

Definition 4.3. Let H be a right M -module and p ∈ B(`2(N))⊗M suchthat H ∼= p · (`2(N) ⊗ L2(M)). The M -dimension of H is defined as thenumber

dim(HM) := (Tr⊗τ)(p) .

The number dim(HM) was already introduced by Murray and von Neu-mann in [30] as the coupling constant of M on H. Note that p belongsto the II∞ factor B(`2(N))⊗B(L2(M)), endowed with the faithful semi-finite normal trace Tr⊗τ . Recall, from Proposition 4.1, that p dependson the choice of an M -linear isometry u such that p = uu∗. Nevertheless,

4.1 Dimension of modules over a II1 factor 65

the number dim(HM) does not depend on the choice of u. Indeed, ifv : H → `2(N)⊗ L2(M) be another M -linear isometry, we have that

(Tr⊗τ)(vv∗) = (Tr⊗τ)((vu∗)(uv∗))= (Tr⊗τ)(uv∗vu∗)= (Tr⊗τ)(u∗u) .

The dimension dim(HM) is a complete invariant of the moduleH. Indeed,the equivalence of projections in the II∞ factor B(`2(N))⊗M easily yields

dim(KM) = dim(HM) ⇐⇒ KM ∼= HM ,

dim(KM) ≤ dim(HM) ⇐⇒ KM is isomorphic with a submodule of HM .

Remark 4.4. LetH be a right M -module over the II1 factor M . Becauseof Proposition 4.2 the commutant of the right action of M on H is givenby

B(H) ∩M ′ = p(B(`2(N))⊗λ(M)

)p .

Since the representation λ is faithful, we identify, as usual, λ(M) and M ,so B(H) ∩M ′ is endowed with the semi-finite faithful normal trace

Tr(·) := (TrB(`2(N)⊗τ)(p · p) ,

and we have thatdim(HM) = Tr(1) .

We have analogous results for left M -modules that we view as right Mop-module. So every left M -module H is a projective module

HM∼=(`2(N)⊗ L2(M)

)· p

and we havedim(MH) := (Tr⊗τ)(p) .

We state now a series of useful formulas concerning the right moduledimension. One obtains symmetric results for left modules.

Theorem 4.5. Let H be a right M -module. Then,

• dim(HM) <∞⇐⇒M ′ ∩ B(H) is a II1 factor.

• dim((⊕

iHi)M)

=∑

i dim((Hi)M

).


• dim((pL2(M))M

)= τ(p), for every projection p ∈M .

Suppose now that dim(HM) <∞. Then,

• dim((Hp)pMp

)= τM (p)−1 dim(HM), for every projection p ∈M .

• dim((Hp)pM

)= τM ′(p) dim(HM), for every projection p ∈M ′.

• dim(HM) =(dim(HM ′)

)−1.

4.2 The Jones index of a subfactor

Once again this section does not go deep into subfactor theory, the aim isto present the needed material for Chapter 10 and we refer to Jones’ orig-inal paper [24] for proofs and further details. In Chapter 10 we computethe entire bimodule category of a II1 factor M and, as a consequence, thisallows us to compute all finite index subfactors of M , using Jones’ tun-nel construction (see [24, Corollary 3.1.9]). The category of finite indexbimodules over a II1 factor is defined and analyzed in section Chapter 5and Chapter 6.

Definition 4.6. Let N ⊂ M be an inclusion of type II1 factors. Thenumber

[M : N ] := dim(L2(M)N)

is called the Jones index of N in M .

The following example justifies the terminology of index of an inclusionof factors.

Example 4.7. Let H and G be ICC groups where H is a subgroup ofG. We write the coset decomposition G =

⊔i∈I giH. Then, `2(G) =⊕

i∈I `2(giH). The Hilbert space `2(G) is naturally a right L(H)-module.

Since `2(giH) is isomorphic to `2(H) as right L(H)-module, Theorem 4.5implies that

[L(G) : L(H)] =∑i∈I

dim(`2(G)L(H)) = |I| = [G : H] .

4.2 The Jones index of a subfactor 67

Lemma 4.8. Let N ⊂ M be an inclusion of II1 factors and H a rightM -module such that that dim(HM) <∞. Then,

dim(HN) = [M : N ] dim(HM) .

Proposition 4.9. Let N ⊂M ⊂ P be inclusions of II1 factors. Then

• [M : N ] ≥ 1.

• [M : N ] = 1 ⇐⇒M = N .

• [P : N ] = [P : M ][M : P ].

• [M : N ] = [N ′ : M ′].

Proposition 4.10. Let N ⊂M be an inclusion of II1 factors. Let p ∈ Nand q ∈ N ′ ∩M be projections. Suppose that the inclusion N ⊂ M hasfinite index. Then,

• [pMp : pNp] = [M : N ].

• [qMq : Nq] = [M : N ]τM (q)τN ′(q).

Remark 4.11. One can also define the dimension of left and right mod-ule over a tracial von Neumann algebra (M, τ) which may no longer befactorial. There is a notion of index, for non-factorial inclusions. In thecase of non-factorial von Neumann algebras, the dimension depends onthe trace τ . In this thesis we mainly consider modules over II1 factorsand in the rare occasions where we deal with modules over finite vonNeumann algebras, or with non-factorial inclusions, there will always bean obvious ambient trace.

For every finite index inclusion of II1 factors N ⊂M , the relative commu-tant N ′∩M is finite dimensional (see [24, Corollary 2.2.3]). In this thesis,we will often use the following well known principle; see for example [53,Lemma A.3].

Lemma 4.12. Let (M, τ) be a tracial von Neumann algebra and N ⊂M

a finite index subfactor. Let Q ⊂ N be a von Neumann subalgebra suchthat Q′ ∩N is finite dimensional, then also Q′ ∩M is finite dimensional.


4.3 Basic construction

In this section N ⊂ M denotes an inclusion of tracial von Neumannalgebras. Let τ be a faithful normal trace on M and endow N with therestriction of this trace. Note that here we do not assume factoriality of Nand M . We know that there exists a unique trace preserving conditionalexpectation EN : M → N (Proposition 4.4.23 in [48]). More precisely,EN is characterized by the following identity

τ(xEN (y)) = τ(xy), for all x ∈ N, y ∈M . (4.2)

In particular, EN is a projection of norm 1 and thus extends to a projec-tion eN : L2(M) → L2(N).

Definition 4.13. The von Neumann algebra 〈M, eN 〉 generated by M

and the projection eN in B(L2(M)) is called the basic construction.

We present some of the first elementary properties of the basic construc-tion.

Proposition 4.14. Let J be the canonical anti-unitary on L2(M) ex-tending the ∗-operation. Then we have

• eNxeN = EN (x)eN , for all x ∈M .

• eN commutes with J and 〈M, eN 〉 = JN ′J .

Note that the first item in the foregoing proposition implies that A :=spanxeNy | x, y ∈ M is a ∗-subalgebra of B(L2(M)). It can be proventhat 〈M, eN 〉 = A′′.

Proposition 4.15. Let N ⊂M be an inclusion of tracial von Neumannalgebras. Then,

• 〈M, eN 〉 is a factor if and only if N is a factor.

• 〈M, eN 〉 is a factor of type II1 if and only if N is a finite indexsubfactor of M . In that case, we have [〈M, eN 〉 : M ] = [M : N ].

4.3 Basic construction 69

Let N ⊂ M be a finite index subfactor. Denote by Tr the trace on theII1 factor 〈M, eN 〉, it can be proven that

Tr(xeN ) = [M : N ]−1τ(x), for all x ∈M . (4.3)

The following theorem proves that the inclusion M ⊂ 〈M, eN 〉 is themodel for every finite index inclusions of II1 factors.

Theorem 4.16. Let N be a finite index subfactor of the II1 factor M .Then, there exists a subfactor N1 ⊂ N and a projection e0 ∈M such thatM = 〈N, e0〉 and N1 ⊂ N ⊂ M is isomorphic to the basic constructionN1 ⊂ N ⊂ 〈N, eN1〉 with an isomorphism sending e0 to eN1.

This construction is called the tunnel construction and is unique up toconjugation with a unitary in N in the following sense. If P,Q ⊂ N aresubfactors such that P ⊂ N ⊂ M and Q ⊂ N ⊂ M are basic construc-tions, there exists a unitary u ∈ N such that uPu∗ = Q.

Chapter 5

Finite index bimodules

In the previous chapter we have seen that all right N -modules H on theII1 factor N are classified by a number between 0 an +∞ called the rightN -dimension of H. Things change radically when H is also endowed witha commuting left M -module action of a II1 factor M . Such Hilbert spaceH is called an M -N -bimodule, see Definition 2.8. We are interested inM -N -bimodules that have finite left M -module dimension and right N -module dimension. Such bimodules are said to be of finite Jones index.In opposition with modules of finite right-dimension, it is very hard todescribe all finite index bimodules over a given II1 factor. The aim ofthis chapter is to give computational tools to analyze finite index M -M -bimodules over the II1 factor M .

Every finite index M -N -bimodule H is isomorphic with the generic M -N -bimodule

ψ(M)

(p(Cn ⊗ L2(N)

))N , (5.1)

where ψ : M → pNnp is a finite index inclusion. The bimodule actionsare given by

a · ξ · b := ψ(a)ξb, for every a ∈M, b ∈ N, ξ ∈ p(Cn ⊗ L2(N)

).

The left and right dimensions of H can be computed in terms of thetrace of the projection p and this index of ψ(M) ⊂ pNnp. See Propo-sition 5.1. Although the isomorphism (5.1) is easy obtain, it provides auseful tool to analyze finite index bimodules. This chapter is a toolboxin which we express all the material gathered in the previous chapters

72 Chapter 5. Finite index bimodules

(direct sums, Connes tensor product...) in terms of these finite index in-clusions. Throughout this chapter, N , M and P denote type II1 factors,unless explicitly mentioned.

Proposition 5.1. Let M,N be type II1 factors and H an M -N -bimodule.

• If dim(HN ) < ∞ then, there exists a unital ∗-homomorphism ψ :M → pNnp such that

MHN ∼= ψ(M)

(p(Cn ⊗ L2(N)

))N .

• Moreover, if dim(MH) <∞ the inclusion ψ(M) ⊂ pNnp has finiteindex and we have that

dim(HN) = (Tr⊗τ)(p), and dim(MH) =[pNnp : ψ(M)]

(Tr⊗τ)(p),

where Tr denotes the non-normalized trace on Mn(C).

Proof. Proposition 4.2 yields a projection p0 ∈ B(`2(N))⊗N such that(Tr⊗τ)(p0) <∞ and HN

∼= p0

(`2(N)⊗L2(N)

). Denote t := (Tr⊗τ)(p0).

Take n ≥ t and p ∈ Mn(C)⊗N with (tr⊗τ)(p) = tn , where tr denotes the

normalized trace on Mn(C). Using formulas from Theorem 4.5 we get

dim(p(Cn ⊗ L2(N)

)N

)= dim

(pL2(Nn)N

)× (tr⊗τ)(p) = t .

There exists then a right N -module isomorphism u : p(Cn⊗L2(N)

)→ H.

We denote by πr and πl the left and right commuting module actions onH and set

ψ : M → B(p(Cn ⊗ L2(N)

)): x 7→ u∗πl(x)u .

Claim. ψ(M) ⊂ pNnp.

The Hilbert space p(Cn⊗L2(N)

)is turned into an N -N -bimodule by the

following actions:

x · ξ := ρleft(x)ξ = ψ(x)ξ, for all x ∈ N ,

ξ · x := ρright(x)ξ = p(1⊗ Jx∗J)pξ, for all x ∈ N .

5.1 Direct sums 73

The unitary u intertwines the right module actions: uρright(x) = πr(x)u.Then, the map ψ commutes with the right module action ρright and sowe have that ψ(M) ⊂ (p(1⊗JNJ)p)′ = p(Mn(C)⊗N)p. This proves theclaim and we get an isomorphism of M -N -bimodules, which prove thefirst assertion of the proposition.

Suppose that H has finite left M -dimension, we prove that the inclusionψ(M) ⊂ pNnp has finite index. By Lemma 4.8, we have that

[pNnp : ψ(M)] =dim

(ψ(M)p

(Cn ⊗ L2(N)

))dim

(pNnp(p

(Cn ⊗ L2(N)

))= (Tr⊗τ)(p) dim(MH) <∞ .

We introduce now an important notation, constantly used throughout thetext.

Notation 5.2. Let N,M be type II1 factors and ψ : N → pMnp a unital∗-homomorphism. We denote by H(ψ) the Hilbert space(

(Cn)∗ ⊗ L2(M))p ,

endowed with the following M,N -actions:

a · ξ · b := aξψ(b), for all a ∈M, b ∈ N .

We always refer to this bimodule when using the notationH(ψ). However,depending on the situation, we might prefer to work with its left analogue

ψ(M)

(p(Cn ⊗ L2(N)

))N , (5.2)

with a finite index inclusion ψ : M → pNnp.

5.1 Direct sums

Let ψ : N → pMnp and ρ : N → qMmq be finite index inclusions. Wedefine

(ψ ⊕ ρ)(·) :=(ψ(·) 00 ρ(·)

): N → (p⊕ q)Mn+m(p⊕ q) .


Then, we immediately have that

H(ψ)⊕H(ρ) ∼= H(ψ ⊕ ρ) .

5.2 Connes’ tensor product

5.2.1 More on bounded vectors, Pimsner-Popa basis

Let (M, τ) be a II1 factor and N ⊂M an inclusion of II1 factors. In [33,Proposition 1.3] Pimsner and Popa consider M as a right N -module, withthe N -valued inner product given by 〈a, b〉 := EN (a∗b) and prove that Mis finitely generated as right N -module if and only if N ⊂ M has finiteindex. They obtain the following

Proposition 5.3. Let (M, τ) be a II1 factor and N ⊂ M a finite indexinclusion of II1 factors. There exist v1, . . . , vn ∈M and q a projection inN satisfying

EN (v∗i vj) =

0 if i 6= j

1 if i = j and i, j 6= n

q if i = j = n

and such that

x =n∑i=1

vi EN (v∗i x), for all x ∈M .

Alternatively, 1 =∑n

i=1 vieNv∗i and hence [M : N ] = (n− 1) + τ(q).

Proof. Take an integer n ≥ [M : N ], a projection p ∈ Mn(C) ⊗ N suchthat (tr⊗τ)(p) = [M :N ]

n , where tr denotes the normalized trace on Mn(C),and an N -linear unitary

u : p(Cn ⊗ L2(N)

)N → L2(M)N

Since replacing p by an equivalent projection in Mn(C) ⊗ N does notaffect the dimension, we may assume that

p =

1

. . .1

q

.

5.2 Connes’ tensor product 75

Define vi := u(p(ei ⊗ 1)), for all i = 1, . . . , n and

v :=n∑i=1

e∗i ⊗ vi ∈ (Cn)∗ ⊗ L2(M) .

As in the proof of Proposition 5.1 we also have a unital ∗-homomorphismψ : N → pNnp such that

u(ψ(a)ξ

)= au(ξ), for all ξ ∈ p

(Cn ⊗ L2(N)

).

Then, an easy computation proves that av = vψ(a), for all a ∈ N . Itfollows that vv∗ is an element of L1(M) commuting withN . SinceN ⊂M

has finite index, the relative commutant N ′∩M is finite dimensional andthus vv∗ ∈M . In particular, v is bounded and thus vi ∈M , for all i; seesection 2.5. Now, we have that for every ξ, η ∈ p

(Cn ⊗N

),

EN(u(ξ)∗u(η)

)= 〈u(ξ), u(η)〉N= 〈ξ, η〉N , by Lemma 3.22

= ξ∗η .

It follows that EN (v∗i vj) = pij , where pij denotes the (i, j)-matrix coeffi-cient of p. Let η ∈ p

(Cn ⊗ N) and write η = p (

∑ni=1 ei ⊗ ηi). We have

that

n∑i=1

vi EN(v∗i u(η)

)=

n∑i,j=1

vi EN (v∗i vj)ηj

=n−1∑i=1

viηi + vnqηn

=n∑i=1

viηi = u(η) .

This formula implies that 1 =∑n

i=1 vieNv∗i . Applying the trace Tr on

〈M, eN 〉 to this equality yields, by (4.3), that

[M : N ] =n∑i=1

τ(v∗i vi) =n∑i=1

(τ EN )(v∗i vi) = (n− 1) + τ(q) .


We are going to use the Pimsner-Popa basis to prove that for finite indexbimodules, left and right bounded vectors coincide. See also [1, Proposi-tion 1.5]

Lemma 5.4. Let N ⊂ M be an inclusion of finite index II1 factors andMH a left M -module. Then, we have that

0(MH) = 0(NH) .

Proof. We only have to prove that every bounded vector ξ ∈ NH isbounded in MH. Let a ∈ M . Because of Proposition 5.3 we can takeelements v1, . . . , vn ∈ M such that a =

∑ni=1 vi EN (v∗i a). There exists

a constant Ci > 0 such that ‖EN (v∗i a) · ξ‖H ≤ Ci‖EN (v∗i a)‖2. So, weobtain

‖a · ξ‖H ≤n∑i=1

‖vi‖M‖EN (v∗i a) · ξ‖H

≤n∑i=1

‖vi‖MCi‖EN (v∗i a)‖2

=n∑i=1

‖vi‖MCiτ(a∗viv∗i a)1/2

≤

(n∑i=1

‖vi‖2MCi

)‖a‖2 .

Proposition 5.5. Let MHN be a finite index M -N -bimodule. Then, avector ξ ∈ H is left bounded if and only if it is right bounded.

Proof. Write MHN ∼= ψ(M)

(p(Cn ⊗ L2(N))

)N, with ψ : M → pNnp a

finite index inclusion. Then, we have that

0(ψ(M)

(p(Cn ⊗ L2(N))

))= 0

(pNnp

(p(Cn ⊗ L2(N))

), by Lemma 5.4

= p(Cn ⊗N)

=((p(Cn ⊗ L2(N))

)N

)0.

5.2 Connes’ tensor product 77

We give an example of computation of the N -valued inner product fre-quently used throughout this text. A more general formula is proven inLemma 5.7.

Example 5.6. Let N ⊂ M be a finite index inclusion of II1 factorsand H := L2(M) viewed as a right N -module. Because of the foregoingresults, we have HN

0 = N0H = M

0H = M . Then, for every v1, v2 ∈ H0

we have〈v1, v2〉N = EN (v∗1v2) .

We state now a useful lemma providing a formula for the M -valued innerproduct on the set of bounded vectors.

Lemma 5.7. Let ψ : N → pMnp be a finite index inclusion. Then

0H(ψ) = H(ψ)0 =((Cn)∗ ⊗M

)p ,

and for every v1, v2 ∈((Cn)∗ ⊗M

)p,

M〈v1, v2〉 = v2v∗1 and 〈v1, v2〉N = ψ−1

(Eψ(N)(v

∗1v2)

).

Proof. We denote by τ the normalized trace on pMnp. The first assertionis now clear, by Proposition 5.5. Let v1, v2 ∈

((Cn)∗ ⊗ M

)p. Then

Rvi(a) = avi, for i = 1, 2 and every a ∈M . The formula M〈v1, v2〉 = v2v∗1

immediately follows. We also have Lvi(a) = viψ(a), for i = 1, 2 and everya ∈ N . We prove the formula for the right inner product. By uniquenessof the trace preserving conditional expectation, we only have to provethat for all b ∈ N ,

τ(ψ(〈v1, v2〉M )ψ(b)

)= τ

(v∗1v2ψ(b)

).

We compute

τ(ψ(〈v1, v2〉M )ψ(b)

)= (τ ψ)(〈v1, v2〉Mb)= τ(〈v1, v2〉Mb)= 〈1, L∗v1Lv2(b)〉= 〈v1, v2ψ(b)〉 .

Note that the foregoing lemma implies in particular that τ(〈x, y〉N ) =〈x, y〉, for every x, y ∈

((Cn)∗⊗M

)p. This formula was already obtained

in Lemma 3.7.


5.2.2 Basic formulas

Proposition 5.8. Let ψ : N → pMnp be a finite index inclusion, PKMa P -M -bimodule and MHP an M -P -bimodule. Then we have that

PK ⊗M H(ψ)N ∼= P

(((Cn)∗ ⊗K

)p)ψ(N) ,

and

ψ(N)

(p(Cn ⊗ L2(M))

)⊗M HP

∼= ψ(N)

(p(Cn ⊗H)

)P .

Proof. The map

T : K ⊗((Cn)∗ ⊗M)

)p→

((Cn)∗ ⊗K

)p : ξ ⊗ x 7→ ξx

extends to a P -N -bimodular unitary since for every x, y ∈((Cn)∗⊗M)

)p

we have that M〈x, y〉 = yx∗. The second assertion of the propositionfollows analogously.

The previous proposition also implies the following. If ψ : N → pMnp

and ρ : M → qPmq are finite index inclusions then

H(ψ)⊗M H(ρ) ∼= H((id⊗ ψ)ρ) .

Similarly, if

H := ψ(N)

(p(Cn ⊗ L2(M)

))M and K := ρ(M)

(q(Cm ⊗ L2(P )

))P .

Then,

NH⊗M KP ∼= (id⊗ ρ)ψ(N)

((id⊗ ρ)(p)

(Cmn ⊗ L2(P )

))P .

The next corollary also follows immediately from Proposition 5.8.

Corollary 5.9. Let α and β be automorphisms of the von Neumannalgebra M . We have then:

H(α)⊗M H(β) ∼= H(α β) .

5.3 Intertwiners 79

5.3 Intertwiners

The following remark will be frequently used in the sequel.

Remark 5.10. Let N ⊂M be an irreducible inclusion of type II1 factors.Consider a finite index inclusion ψ : M → pMnp. Then, because ofLemma 4.12, we have that

dim(ψ(N)′ ∩ pMnp

)< +∞ .

5.3.1 General facts

Proposition 5.11. Let M,N be II1 factors. Consider the unital ∗-homomorphism ψ : M → pNnp and an M -N -bimodule H. WheneverT is an M -N -bimodular map

T : ψ(M)

(p(Cn ⊗ L2(N))

)N → MHN ,

there exists a vector ξ ∈ ((Cn)∗ ⊗ H)p such that a · ξ = ξ · ψ(a), for alla ∈M and such that

T (x) = ξx, for every x ∈ p(Cn ⊗N) .

Proof. Define ξ ∈((Cn)∗ ⊗H

)p by the formula

ξ =n∑i=1

T (p(ei ⊗ 1))(e∗i ⊗ 1) .

It follows that a · ξ = ξ · ψ(a) for all a ∈M . Indeed

a · ξ =n∑j=1

T (pψ(a)(ej ⊗ 1))(e∗j ⊗ 1), by left-M -linearity of T

=n∑

i,j=1

T (p(ei ⊗ ψ(a)ij))(e∗j ⊗ 1)

=n∑

i,j=1

T (p(ei ⊗ 1))(e∗j ⊗ ψij(a)), by right-N -linearity of T

=n∑i=1

T (p(ei ⊗ 1))(e∗i ⊗ 1)n∑

k,l=1

eke∗l ⊗ ψ(a)kl

= ξ · ψ(a) .


We prove now the second assertion. Let x :=∑n

i=1 p(ei⊗xi) ∈ p(Cn⊗N).

T (x) =n∑i=1

T (p(ei ⊗ 1))(1⊗ xi)

=n∑

i,j=1

T (p(ei ⊗ 1))(e∗i ⊗ 1)(ej ⊗ xj)

= ξ · x, since ξ = ξp .

Remark 5.12. Note that by construction of the vector ξ, the vectorspace [ξ · p(Cn ⊗N)] is an M -N -bimodule. It is isomorphic with H whenT has dense range.

In the previous proposition, we wrote the generic M -N -bimodule with aleft action given by the ∗-homomorphism ψ : M → pNnp. For conve-nience, we briefly state the right analogue. Let H be an N -M -bimoduleand T ∈ MorN

(H(ψ),H

)M

. We define the vector

ξ :=n∑i=1

(ei ⊗ 1)T ((e∗i ⊗ 1)p) ∈ p(Cn ⊗H) (5.3)

and obtain the right analogue of Proposition 5.11. Indeed, similar com-putations yield that ψ(a) · ξ = ξ · a, for all a ∈M and T (x) = xξ, for allx ∈

((Cn)∗ ⊗N)

)p.

We can now derive many little corollaries from the foregoing easy propo-sition.

Corollary 5.13. Let A,B,N,M be II1 factors such that A ⊂ B andN ⊂ M . Consider the unital ∗-homomorphisms ψ : A → pNnp andη : B → qMmq and suppose that

• A ⊂ B is irreducible,

• η(B) ⊂ qMmq has finite index

Every A-N -bimodular bounded operator

T : ψ(A)

(p(Cn ⊗ L2(N)

))N → η(A)

(q(Cm ⊗ L2(M)

))N

5.3 Intertwiners 81

is given by left multiplication by an element of

L := v ∈ q(Cm(Cn)∗ ⊗M)p | η(a)v = vψ(a) for all a ∈ A .

Proof. Let T : p(Cn ⊗ L2(N)) → q(Cm ⊗ L2(M)) be a bounded A-N -bimodular operator. Proposition 5.11 yields an element ξ ∈ q(Cm(Cn)∗⊗L2(M))p such that η(a)ξ = ξψ(a) for all a ∈ A and T (x) = ξx, forall x ∈ p(Cn ⊗ N). Then, as an element of q

(Mm(C) ⊗ L1(M)

)q, the

operator ξξ∗ belongs to η(A)′ ∩ qMmq (see section 2.5). The later is afinite dimensional algebra by Remark 5.10. So the element ξ is boundedand thus belongs to L.

Note that in the case where η = ψ, we obtain:

Corollary 5.14. Let ψ : N → pMnp be finite index inclusion. Then, wehave that

MorM (H(ψ))N ∼= ψ(N)′ ∩ pMnp .

Remark that from this corollary, we obtain in particular that H(ψ) isirreducible if and only if the inclusion ψ(N) ⊂ pMnp is irreducible.

Proposition 5.15. Let ψ : N → pMnp and η : N → qMmq be finiteindex inclusions. Then:

1. H(η) is isomorphic to a subbimodule of H(ψ) ⇔ there exists a par-tial isometry v ∈ p(Cn(Cm)∗ ⊗M)q satisfying vη(x) = ψ(x)v, forall x ∈ N and vv∗ = p.

2. H(ψ) ∼= H(η) ⇔ there exists an element u ∈ p(Cn(Cm)∗ ⊗ M)qsatisfying uη(x) = ψ(x)u, for all x ∈ N and uu∗ = p, u∗u = q.

Proof. Point 1 follows from Corollary 5.13. Indeed, H(η) ⊂ H(ψ) yieldsa projection T ∈ MorM (H(ψ))N such that H(η) = Im(T ). Becauseof Corollary 5.13, the map T is the right multiplication by an elementv ∈ p(Cn(Cm)∗ ⊗M)q. This element v is the required partial isometryand satisfies T = vv∗.

Point 2. The isomorphism H(ψ) ∼= H(η) yields an M -N -bimodularunitary w ∈ MorM (H(ψ),H(η))N . Corollary 5.13 provides an elementu ∈ p(Cn(Cm)∗ ⊗ M)q such that w = Ru, where Ruξ := ξu for allξ ∈

((Cn)∗ ⊗M

)p. The fact that w is a unitary and that u 7→ Ru is

injective easily imply that u is a unitary.


Lemma 5.16. Let N ⊂M be an irreducible inclusion of II1 factors andMKM a finite index M -M -bimodule. Whenever NHN is a finite index N -N -bimodule, the vector space of bounded N -N -bimodular operators fromH to K, is finite dimensional.

Proof. Consider the finite index unital ∗-homomorphisms ψ : M → pMnp

and η : N → qNmq such that

H ∼= ψ(M)

(p(Cn ⊗ L2(M)

))M and K ∼= η(N)

(q(Cm ⊗ L2(M)

))M .

Corollary 5.13 implies that the space of intertwiners MorN (H,K)N isisomorphic to

L = v ∈ q(Cm(Cn)∗ ⊗M)p | η(a)v = vψ(a) for all a ∈ N .

Then, we are left with proving that L is finite dimensional. Define A :=pMnp∩ψ(N)′. Since N ⊂M is irreducible and ψ(M) ⊂ pMnp has finiteindex, the algebra A is finite dimensional (see Lemma 4.12). Let p1, . . . , prbe a maximal set of mutually orthogonal minimal projections in A forwhich there exist vi ∈ L with v∗i vi = pi. For every i, let qi1, . . . , qisi be amaximal set of mutually orthogonal projections in qMmq for which thereexist vij ∈ L satisfying v∗ijvij = pi and vijv∗ij = qij . Since Tr(qij) = Tr(pi)for every j, it follows that si <∞. It is now easy to check that

L = spanvij | i = 1, . . . , r, j = 1, . . . , si .

5.3.2 Connes tensor product versus product in a givenmodule

Lemma 5.17. Let N ⊂ M be an irreducible inclusion of II1 factors.Suppose that K ⊂ L2(M) is an N -N -subbimodule of finite index.

1. Choose a projection p ∈ Nn, a finite index inclusion ϕ : N → pNnp

and an N -N -bimodular unitary

U : ϕ(N)p(Cn ⊗ L2(N))N → NKN .

5.3 Intertwiners 83

Then, U(p(Cn ⊗N)

)= K ∩M and defining v ∈ (Cn)∗ ⊗M by the

formula

v :=n∑i=1

e∗i ⊗ vi with vi := U(p(ei ⊗ 1)) ,

we have vp = v, av = vϕ(a) for all a ∈ N and U(ξ) = vξ for allξ ∈ p(Cn⊗L2(N)). In particular, K∩M = spanviN | i = 1, . . . , nand K ∩M is dense in K.

2. Let P be a II1 factor and MHP an M -P -bimodule. Suppose thatL ⊂ H is a closed N -P -subbimodule. Denote by K ∗ L the closureof (K ∩M)L inside H. Then, K ∗ L is an N -P -bimodule that isisomorphic to a subbimodule of K ⊗N L. Furthermore, wheneverK0 ⊂ K ∩M is such that K0 ⊂ K is dense, also K0L follows densein K ∗ L. If K ∗ L is non-zero and K⊗N L is irreducible, it followsthat K ∗ L and K ⊗N L are isomorphic N -P -bimodules.

By symmetry, similar statements hold on the right. In particular, when-ever PHM is a P -M -bimodule with closed P -N -subbimodule L, we defineL∗K as the closure of L(K∩M) inside H and find that L∗K is isomorphicwith a P -N -subbimodule of L ⊗N K.

Proof. Choose a projection p ∈ Nn, a finite index inclusion ϕ : N →pNnp and an N -N -bimodular unitary

U : ϕ(N)p(Cn ⊗ L2(N))N → NKN .

We apply here exactly the same reasoning as in the proof of Proposi-tion 5.11. Define vi ∈ L2(M) by the formula vi := U(p(ei ⊗ 1)). Putv :=

∑ni=1 e

∗i ⊗ vi, which belongs to (Cn)∗ ⊗ L2(M). By construction,

spanviN | i = 1, . . . , n is dense inK and U(ξ) = vξ for all ξ ∈ p(Cn⊗N).Since U is N -N -bimodular, we have av = vϕ(a) for all a ∈ N and, inparticular, v = vp. It follows that vv∗ is an element of L1(M) commutingwith N . By the irreducibility of N ⊂M , we get vv∗ ∈ C1. In particular,v is bounded and vi ∈M for every i, see section 2.5.

Since U(ξ) = vξ for all ξ ∈ p(Cn ⊗ N), also U∗(b) = (id ⊗ EN )(v∗b)for all b ∈ M . In particular, U∗(M) ⊂ p(Cn ⊗ N) and it follows thatK∩M = U(p(Cn⊗N)) = spanviN | i = 1, . . . , n. This proves the firstpart of the lemma.


To prove the second part, let P be a II1 factor and MHP anM -P -bimodulewith closed N -P -subbimodule L. Define K∗L as the closure of (K∩M)Linside H. Define the subspace L0 ⊂ L of vectors ξ ∈ L such that the mapN → L : a 7→ aξ extends to a bounded operator from L2(N) to L. Then,L0 is dense in L. Fix ξ ∈ L0. We claim that the map R : K ∩M → H :a 7→ aξ extends to a bounded operator from K to H. Since ξ ∈ L0, themap

S : Cn ⊗N → H : ei ⊗ a 7→ viaξ

extends to a bounded operator from Cn ⊗ L2(N) to H, that we stilldenote by S. By construction, S(ϕ(a)η) = aS(η) for all a ∈ N andη ∈ Cn ⊗ L2(N). In particular, S(η) = S(pη) for all η ∈ Cn ⊗ L2(N). Itfollows that R(U(η)) = S(η) for all η ∈ p(Cn ⊗ N). Since U is unitaryand K ∩M equals U(p(Cn ⊗N)), the claim is proven.

Suppose now that K0 ⊂ K∩M and that K0 ⊂ K is dense. From the claimin the previous paragraph, we get

K ∗ L = (K ∩M)L = (K ∩M)L0 = K0L0 = K0L .

So, K0L is dense in K ∗ L.

The Connes tensor product K⊗N L can be realized as the N -P -bimoduleϕ(N)p(Cn ⊗ L)P , by Proposition 5.8. The linear operator T : p(Cn⊗L) →H : T (ξ) = vξ is N -P -bimodular with range spanviL | i = 1, . . . , n.From the results above, it follows that the closure of the range of T equalsK ∗L. Taking the polar decomposition of T ∗, we find an N -P -bimodularisometry of K ∗ L into K ⊗N L.

5.3.3 Intertwining bimodules

In this section, we briefly recall some aspects of Popa’s intertwining-by-bimodules technique introduced in [36]. This very powerful technique isused to deduce unitary conjugacy of two von Neumann subalgebras A andB of a tracial von Neumann algebra (M, τ) from their weak containmentA ≺M B that we define now.

Definition 5.18. Let (M, τ) be a von Neumann algebra with normalfaithful tracial state. Let A,B ⊂ M be unital subalgebras. We writeA ≺M B if L2(M) contains a non-zero A-B-subbimodule K such thatdim(KB) <∞.

5.3 Intertwiners 85

The following theorem is [34, Theorem 2.1].

Theorem 5.19. Let (M, τ) be a von Neumann algebra with normal faith-ful tracial state. Let A,B ⊂ M be unital subalgebras. The following areequivalent.

1. A ≺M B.

2. There exist n ∈ N and

• a unital ∗-homomorphism ψ : A→ pBnp ,

• a non-zero partial isometry v ∈((Cn)∗ ⊗M

)p

satisfyingav = vψ(a) for all a ∈ A . (5.4)

3. There exists a non-zero element in a ∈ 〈M, eB〉+ ∩ A′ such thatTr(a) <∞. The trace Tr denotes the semi-finite trace on the basicconstruction 〈M, eB〉.

4. There exists no sequence of unitaries (un) ∈ A such that

‖EB(aunb)‖2 → 0, for all a, b ∈M .

The assertion 4 is the important statement of the foregoing theorem. Itprovides a useful tool to prove that A ≺M B, reasoning by contradiction.We only give a partial proof of this theorem and refer to [34, Theorem2.1] for a complete proof (see also [55, Proposition C.1]). In this thesis,we never use assertion 4, we only use 2 and 3 which are easy reformula-tions of the fact that L2(M) contains an A-B-subbimodule which is finitedimensional on the right.

The idea of Popa is to deduce, in the good cases, the existence of a unitaryu ∈ M intertwining the subalgebras A and B from the existence of anintertwining bimodule. If there exists such unitary u ∈ U(M) satisfyingu∗Au = B, then, L2(uB) ⊂ L2(M) is an A-B-bimodule with finite rightB-dimension. Usually the existence of an intertwining bimodule is veryfar from implying unitary conjugacy, it only implies that a corner of Acan be unitarily conjugated into a corner of B. In the case of Cartansubalgebras of a II1 factor, Popa proved that these notions are equivalent[36, Theorem A.1]. See also [55, Proposition C.3]


Proof of Theorem 5.19. 2 ⇒ 1. Let ψ : A → pBnp be a unital ∗-homo-morphism and v ∈

((Cn)∗ ⊗M

)p a non-zero partial isometry satisfying

av = vψ(a) for all a ∈ A. Then,

A

(spanv(ei ⊗B) | i = 1, . . . , n

)B

is an A-B-bimodule, finitely generated as right B-module.

1 ⇒ 2. Suppose that there exists a non-zero A-B-subbimodule K withdim(KB) < ∞. Take a central projection z ∈ B such that Kz is finitelygenerated as right B-module (see [55, Lemma A.1] for a proof of thisresult). So, there exists an integer n ∈ N, a projection p ∈ Bn and a rightB-linear isomorphism

T : p(Cn ⊗ L2(B)

)→ Kz .

Denote by πl the left A-action on K. Following exactly the lines of theproof of Proposition 5.1, we find that

ψ(·) := T ∗πl(·)T : A→ pBnp

and we have an isomorphism of A-B-bimodules

A

(Kz)B∼= ψ(A)

((Cn ⊗ L2(B)

)p)B .

Then, T ∈ MorA(p(Cn ⊗ L2(B)

),Kz)B and Proposition 5.11 yields an

element ξ ∈((Cn)∗ ⊗ K

)p such that a · ξ = ξ · ψ(a), for all a ∈ A. Note

that since K ⊂ L2(M), the element ξ belongs to((Cn)∗⊗L2(M)

)p. Then,

the elements

Xa :=(a 00 ψ(a)

)∈Mn+1 and Ξ :=

(0 ξ

0 0

)∈ Mn+1(C)⊗ L2(M) .

satisfy XaΞ = ΞXa, for all a ∈ A. The polar decomposition of Ξ (seesection 2.5) yields a partial isometry

V :=(

0 v

0 0

)satisfying XaV = V Xa, for all a ∈ A. This relation implies that av =vψ(a) and the assertion 2 is proven.

1 ⇒ 3. Let H ⊂ AL2(M)B be a non-zero A-B-subbimodule of L2(M) suchthat dim(HB) < ∞. Let p1 be the projection on H in L2(M). Since H

5.3 Intertwiners 87

is a left A-module, the projection p1 belongs to the commutant of A.Moreover, H being a right B-module, the projection p1 commutes to theright B-action and thus belongs to the basic construction 〈M, eB〉. Theprojection p1 is non-zero since H is non-zero. Viewing 〈M, eB〉 as thecommutant of the right B-module action on H, we have (see Remark 4.4)that TrB(p1) = dim(HB) <∞. Assertion 3 is proven.

3 ⇒ 1. Let a ∈ 〈M, eB〉+ ∩A′ such that 0 < Tr(a) <∞. Take δ > 0 anddefine the spectral projection p := χ]δ,+∞[(a). Then, p ∈ 〈M, eB〉+ ∩ A′,by functional calculus, so K := Im(p) is a non-zero A-B-subbimodule ofL2(M). Viewing again 〈M, eB〉 as the commutant of the right B-moduleaction on K, we have that

dim(KB) = Tr(p) ≤ δ−1 Tr(a) <∞ .

The following technical remark will be often used in the sequel.

Remark 5.20. In Theorem 5.19, we may assume that p is the supportprojection of EB(v∗v), meaning that p is the smallest projection in Bn

such that pEB(v∗v) = EB(v∗v). Let q be a projection in ψ(A)′ ∩ pBnp

such that qEB(v∗v) = EB(v∗v). Then, formula (5.4)implies that

avq = vqψ(a), for all a ∈ A . (5.5)

Denote by w the partial isometry in the polar decomposition of vq and

η : A→ qBnq : a 7→ ψ(a)q .

Formula (5.5) implies that η(a) and (vq)∗(vq) commute, for all a ∈ A.By functional calculus, we obtain that

aw = wη(a), for all a ∈ A .

Note that w is non-zero since EB((vq)∗(vq)) = qEB(v∗v)q = EB(v∗v)and v is non-zero. We know that the polar decomposition of qv∗ is givenby qv∗ = w∗|qv∗|. Then, by uniqueness of the polar decomposition, wefind that qw∗ = w∗. Then qEB(w∗w) = EB(w∗w). If q is the supportprojection of EB(w∗w), we are done and if there exists a projection q′

smaller than q in ψ(A)′ ∩ pBnp such that q′ EB(w∗w) = EB(w∗w), weapply the same procedure. Finally, we may assume from the beginningthat p is the support projection of EB(v∗v).


Intertwining bimodules and non-unital inclusions

In this last section, we extend the notation A ≺M B to non-unital in-clusions of A and B into matrices over M . We have postponed thisgeneralization until now since the main idea is really Theorem 5.19 andthis technical remark will only be needed in the last two chapters.

Definition 5.21. Let (M, τ) be a von Neumann algebra with normalfaithful tracial state. Let A,B ⊂Mn be possibly non-unital embeddings.We write A ≺

MB if 1A(Mn(C) ⊗ L2(M))1B contains a non-zero A-B-

subbimodule K with dim(KB) <∞.

By Theorem 5.19, we have A ≺M B if and only if there existm ∈ N, a non-zero partial isometry v ∈ 1A

(Cn(Cm⊗Cn)∗⊗M

)(1⊗ 1B) and a possibly

non-unital ∗-homomorphism ψ : A → Mm(C) ⊗ B satisfying av = vψ(a)for all a ∈ A. In particular, writing for i = 1, . . . ,m, vi := v(ei ⊗ 1⊗ 1),we have found v1, . . . , vm ∈ 1AMn1B such that spanviB | i = 1, . . . ,mis a left A-module.

5.4 Decomposition into irreducibles

Proposition 5.22. Let ψ : N → pMnp be a finite index inclusion. ThenH(ψ) decomposes as a direct sum of finite index M -N -bimodules

H(ψ) =⊕q∈I

H(ψ(·)q) ,

where I is the set of minimal projections in the finite dimensional alge-bra pMnp ∩ ψ(N)′. The decomposition is unique up to permutation andconjugation by a unitary.

Proof. The algebra pMnp∩ψ(N)′ is finite dimensional, by Remark 5.10.Let p1, . . . , pr be a family of minimal pairwise orthogonal projectionsin pMnp ∩ ψ(N)′ summing to 1. For all i ∈ 1, . . . , r, the bimoduleH(ψ(·)pi) is an irreducible subbimodule of H(ψ) since

(ψ(N)pi)′ ∩ piMnpi = pi(ψ(N)′ ∩ pMnp

)pi = Cpi .

Since we obviously have H(ψ)pi = H(ψ(·)pi), we have the required de-composition.

5.5 Contragredient bimodule 89

5.5 Contragredient bimodule

Lemma 5.23. Let ψ : N → pMnp be a finite index inclusion definingthe finite index bimodule N -M -bimodule

NHM := ψ(N)

(p(Cn ⊗ L2(M)

))M .

Then, the contragredient bimodule of H is given by

ψ(N)

(p(Cn ⊗ L2(M)

))M∼= M

(((Cn)∗ ⊗ L2(M)

)p)ψ(N) .

Proof. Denote by J the canonical anti-unitary on L2(M). Then, the map

T : ψ(N)

(p(Cn ⊗ L2(M)

))M → M

(((Cn)∗ ⊗ L2(M)

)p)ψ(N)

such that T ((ξ1, . . . , ξn)ᵀ) = (Jξ1, . . . , Jξn) is the required M -N -bimodu-lar unitary, as easy matrix computations show.

5.6 Conjugates and Frobenius reciprocity

Throughout this section MHN and NKM are finite index bimodules. Werefer to bounded vectors without specifying left or right since these setsof vectors coincide, by Proposition 5.5. We define, for ξ ∈ H0, η ∈ K0

the operators

λξ : K → H⊗N K : w 7→ ξ ⊗ w ,

ρη : H → H⊗N K : w 7→ w ⊗ η .

Using Lemma 3.8 and formula 3.5, we find that the adjoints are given by

λ∗ξ(a⊗ µ) = 〈ξ, a〉N · µ, for every a ∈ H0, µ ∈ K ,

ρ∗η(µ⊗ a) = µ · N〈η, a〉, for every a ∈ K0, µ ∈ H .

Proposition 5.24. There is a unique element R ∈ MorM (L2(M),H⊗NH)M such that, for all ξ ∈ H0,

• λ∗ξ(R(1)) = ξ ,

• ρ∗ξ(R(1)) = ξ .


Proof. Let ψ : M → pNnp be a finite inclusion such that we have thefollowing M -M -bimodular unitary

θ : H → ψ(M)p(Cn ⊗ L2(N)

)N .

The map θ implements a unitary between the contragredient bimodules.Then, the M -bimodular isometry

φ : H0 ⊗N H0 → ψ(M)

(p(Mn(C)⊗N

)p)ψ(M) : ξ ⊗ η 7→ θ(ξ)θ(η)

extends to a unitary MH⊗N HM ∼= ψ(M)

(p(Mn(C)⊗ L2(N)

)p)ψ(M). So,

the element φ−1(p) is an M -central vector in H⊗N H. Then, we have anisomorphism L2(M) ∼= M · φ−1(p). So we get a map

R : L2(M) → H⊗N H : a 7→ a · φ−1(p) .

We easily obtain the formulas stated in the proposition. For every ξ, η ∈H0

〈λξ(η), φ−1(p)〉 = 〈φ(λξ(η)), p〉= 〈θ(ξ)θ(η), p〉= 〈θ(η), θ(ξ)〉= 〈η, ξ〉 .

So, λ∗ξ(R(1)) = ξ. The formula ρ∗ξ(R(1)) = ξ is obtained similarly.

Notation 5.25. We denote by 1H the identity map on H. When noambiguity is possible, we drop the subscript H and simply write 1.

In the proof of the foregoing lemma, we chose not to make any iden-tification and to write explicitly all unitary operators between equiv-alent bimodules. Since unitary operators preserve norms and in or-der to carry lighter notations, we will consider, from now on, unitaryequivalent bimodules as really equal. For example, we identify H andL2(M) ⊗M H, so if S is a morphism H → H, we also identify the mor-phism 1L2(M) ⊗M S : L2(M)⊗M H → L2(M)⊗H with S.

Proposition 5.26. Denote by RH ∈ MorM (L2(M),H⊗NH)M and RH ∈MorN (L2(N),H⊗M H)M the canonical morphisms obtained in the previ-ous proposition. Then, they satisfy

(R∗H ⊗M 1H)(1H ⊗N RH) = 1 and ,

(1H ⊗M R∗H)(RH ⊗N 1H) = 1 .

5.6 Conjugates and Frobenius reciprocity 91

The notion of tensor product intertwiners was defined in Proposition 3.23.Before starting the proof, note that we have

1H ⊗N RH : H⊗N L2(N) → (H⊗N H)⊗M H, and

R∗H ⊗M 1H : H⊗N (H⊗M H) → L2(M)⊗M H .

So in particular

(R∗H ⊗M 1H)(1H ⊗N RH) : H⊗N L2(N) → L2(M)⊗M H .

Proof. In order to make the following computations easier to read, werecall the needed formulas, although they are all contained in Proposition5.24 and the foregoing definitions. For every η ∈ H0, we have that

ρη : H → H⊗N H : µ 7→ µ⊗ η, and ρ∗η(RH(1)) = η ,

ρη : H → H⊗M H : µ 7→ µ⊗ η, and ρ∗η(RH(1)) = η ,

Under the identifications H⊗N L2(N) ∼= H and that L2(M)⊗M H ∼= H,we have, for every ξ, η ∈ H0,

〈(R∗H ⊗M 1H)(1H ⊗N RH)ξ, η〉 = 〈ξ ⊗RH(1), RH(1)⊗ η〉= 〈ξ ⊗RH(1), (idH ⊗ ρη)(RH(1))〉= 〈ξ ⊗ ρ∗η(RH(1)), RH(1)〉= 〈ξ ⊗ η,RH(1)〉= 〈ρη(ξ), RH(1)〉= 〈ξ, ρ∗η(RH(1))〉= 〈ξ, η〉 .

The second formula is obtained in a similar way, using the map λξ : H →H⊗M H.

As a corollary, we obtain the important Frobenius reciprocity theorem.Before starting the proof, we give some useful formulas.

Lemma 5.27. Let H and K be respectively finite index M -N and N -P -bimodules. We have that

• 〈RH(1), ξ ⊗ η〉 = 〈η, ξ〉, for every ξ, η ∈ H0 ,

• 〈RH(1), ξ ⊗ η〉 = 〈ξ, η〉, for every ξ, η ∈ H0 ,


• 〈(RH ⊗ 1K)η, ξ ⊗ µ〉 = 〈ξ ⊗ η, µ〉, for every ξ ∈ H0, η ∈ K0, µ ∈H0 ⊗N K0.

Proof.

〈RH(1), ξ ⊗ η〉 = 〈λ∗ξ(RH(1)), η〉= 〈ξ, η〉= 〈η, ξ〉 .

The second formula is obtained by exchanging H and H. We prove nowthe last formula.

〈(RH ⊗ 1K)η, ξ1 ⊗ ξ2 ⊗ ξ3〉 = 〈RH(1)⊗ η, ξ1 ⊗ ξ2 ⊗ ξ3〉= 〈RH(1), ξ1 ⊗ ξ2〉〈η, ξ3〉= 〈ξ1, ξ2〉〈η, ξ3〉= 〈ξ1 ⊗ η, ξ2 ⊗ ξ3〉 .

which allows us to conclude.

Theorem 5.28. Let H be an M -N -bimodule, K an N -P -bimodule andL an M -P -bimodule. We also assume that H, K and L all have finiteindex. Then, the maps

MorM (H⊗N K,L)P → MorN (K,H⊗M L)P : T 7→ (1H ⊗ T )(RH ⊗ 1K)

MorM (H⊗N K,L)P → MorM (H,L ⊗P K)N : T 7→ (T ⊗ 1K)(1H ⊗RK)

are natural isomorphisms.

Proof. We only prove that the first map Θ : T 7→ (1H ⊗ T )(RH ⊗ 1K) isa natural isomorphism, the other one follows similarly. The inverse of Θis given by Ψ(S) = (R∗H ⊗ 1L)(1H ⊗ S). We compute

Θ(Ψ(S)

)=

(1H ⊗

((R∗H ⊗ 1L)(1H ⊗ S)

))(RH ⊗ 1K

)= (1H ⊗R∗H ⊗ 1L)(1H ⊗ 1H ⊗ S)(RH ⊗ 1K)

= (1H ⊗R∗H ⊗ 1L)(RH ⊗ 1H ⊗ 1L)(1L2(N) ⊗ S)

= S, by Proposition 5.26 .

Similarly, Ψ(Θ(S)

)= S. The isomorphism Θ is natural in H,K and L,

see appendix for the details of the computations.

5.7 Dimension function 93

Definition 5.29. Let H,K,L be M -N , N -P and M -P finite index andirreducible bimodules. We define the numbers

m(H,K;L) := dim(MorM (L,H⊗M K)P

).

When H and K are finite index irreducible bimodules, the tensor productH⊗M K may be no longer irreducible. We have seen in Proposition 5.22that H⊗M K splits in a direct sum of irreducibles. The map p 7→ Im(p)is a bijection between the set of projections in MorM (H ⊗N K)P andthe set of M -P -subbimodules of H⊗N ⊗K. Moreover, isomorphic M -P -subbimodules correspond to equivalent projections in MorM (H⊗N K)P .Then, every irreducible M -P -subbimodule L of H ⊗N K appear exactlywith multiplicity equal to dim

(MorM (L,H⊗M K)P

)so we have that

H⊗M K ∼=⊕Lm(H,K;L) L .

Because of Frobenius reciprocity Theorem 5.28, we can prove that thesenumbers satisfy the following formulas.

Proposition 5.30. Let H,K,L be respectively finite index irreducible M -N , N -P and P -Q-bimodule. Then we have that

m(H,K;L) = m(H,L;K) = m(L,K;H) .

5.7 Dimension function

The pair (RH, RH) satisfying the relation of Proposition 5.26 is called apair of conjugates, in the language of C∗-tensor categories; see for example[12] and section 6.2. In [46], J.E. Roberts defined a notion of dimensionof an object possessing a pair of conjugates in a C∗-tensor category. Seealso Longo and Roberts theory of dimension developed in [26]. FollowingRoberts’ theory and using the pair of conjugates (RH, RH), they con-struct a self-adjoint multiplicative and additive dimension function onthe class of finite index M -N -bimodules that we reproduce here. We usethe notations of the previous sections.

Proposition 5.31. For every finite index M -N -bimodule H, the linearmaps

τ lH : MorM (H)N → C : τ lH(T )1 = R∗H(T ⊗ 1)RH and

τ rH : MorM (H)N → C : τ rH(T )1 = R∗H(1⊗ T )RH ,


define faithful traces on MorM (H)N . Denote by QH ∈ MorM (H)N theunique element such that τ lH(T ) = τ rH(TQH), for all T ∈ MorM (H)N .Then, the element QH satisfy the following properties.

• QH is positive, invertible and belongs to the center of MorM (H)N .

• QH⊗NK = QH ⊗QK, for every finite index N -P -bimodule K.

• QH = (QH)−1.

• If vi ∈ MorM (Hi,H)N such that v∗i vi = 1 and∑n

i=1 viv∗i = 1 then,

QH =∑n

i=1 viQHiv∗i .

Proof. Note that R∗H(T ⊗ 1)RH ∈ MorM (L2(M))M = C1. Let ψ : M →pNnp be a finite index inclusion such that MHN ∼= ψ(M)p(Cn ⊗ L2(N))N.Then, we have that MorM (H)N = ψ(M)′ ∩ pNnp. Through this iso-morphism, the map RH is identified with ψ and the element T ⊗ 1 ∈MorM (H ⊗ H)M with the operator MT of left multiplication by T ∈ψ(M)′ ∩ pNnp. Since R∗H(T ⊗ 1)RH ∈ C1, we have, in particular,(R∗H(T ⊗ 1)

)(RH(1)

)∈ C1 and thus,

τ lH(T ) = τ(ψ∗(Tp)

)= τ(ψ∗(T ))

= (Tr⊗τ)(T ) .

This proves that τ lH is a faithful trace. Similar computations can becarried for τ rH. We construct now a positive and invertible element QH ∈MorM (H)N such that τ lH(T ) = τ rH(TQH), for every T ∈ MorM (H)N . Weknow that ψ(M)′ ∩ pNnp is a finite dimensional C∗-algebra, by Lemma4.12, and thus, there exist integers n1, . . . , nk such that

MorM (H)N ∼=k⊕i=1

Mni(C) ,

see for example [52, Theorem 11.2] for the structure of finite dimensionalC∗-algebras. We denote by τn the only normalized trace on Mn(C). Then,a trace tr on MorM (H)N is the data of a k-tuple

α := (α1, . . . , αk) ∈ Rk+ such that

k∑i=1

αi = 1, satisfying tr =k∑i=1

αiτi .


Take such k-tuples α and β defining, respectively, the trace τ lH and τ rH.Then, the element

QH :=k⊕i=1

αiβi

idMni (C)

has the required properties.

We prove now that QH⊗NK = QH ⊗QK. We start with proving that

RH⊗NK = (1H ⊗RK ⊗ 1H)RH . (5.6)

By M -linearity of the maps RH and RK, it is only necessary to prove thatRH⊗NK and (1H ⊗RK ⊗ 1H)RH coincide in 1. We have

1H ⊗RK ⊗ 1H : H⊗H → H⊗N (K ⊗P K)⊗N H ,

and write

ξ := ξ1 ⊗ ξ2 ∈ H0 ⊗N K0, η := η2 ⊗ η1 ∈ H0 ⊗N K0 .

Denote by a := R∗K(ξ2⊗η1) ∈ N . Then, using the first formula of Lemma5.27, we obtain

〈(1H ⊗RK⊗1H)RH(1), ξ ⊗ η〉= 〈RH(1), ξ1 ⊗ a⊗ η2〉= 〈RH(1), ξ1a⊗ η2〉, identifying H ∼= H⊗N L2(N)

= 〈η2, ξ1a〉= 〈η2, ξ1〉〈1, a〉, via the same identification

= 〈η2, ξ1〉〈RK(1), ξ2 ⊗ η1〉= 〈η2, ξ1〉〈η1, ξ2〉= 〈η, ξ〉= 〈RH⊗NK(1), ξ ⊗ η〉 .


Then, we compute, for all T ∈ MorM (H⊗N K)P ,

τ lH⊗NK(T )

= R∗H⊗NK(T ⊗ 1K ⊗ 1H)RH⊗NK

= R∗H(1H ⊗R∗K ⊗ 1H)(T ⊗ 1K ⊗ 1H)(1H ⊗RK ⊗ 1H)RH= R∗H

((1H ⊗R∗K)(T ⊗ 1K)(1H ⊗RK)⊗ 1H

)RH

= τ lH((1H ⊗R∗K)(T ⊗ 1K)(1H ⊗RK)

)= τ rH

((1H ⊗R∗K)(T ⊗ 1K)(1H ⊗RK)QH

)= R∗H

(1H ⊗

((1H ⊗R∗K)(T ⊗ 1K)(1H ⊗RK)QH

))RH

= R∗K

(R∗H ⊗ 1K ⊗ 1K)(1H ⊗ T ⊗ 1K)

((1H ⊗QH)RH ⊗ 1K ⊗ 1K

))RK

= τ lK

(R∗H ⊗ 1K)(1H ⊗ T )

((1H ⊗QH)RH ⊗ 1K

))= τ rK

(R∗H ⊗ 1K)(1H ⊗ T )

((1H ⊗QH)RH ⊗ 1K

)QK

)= R∗K

(1K ⊗

((R∗H ⊗ 1K)(1H ⊗ T )

((1H ⊗QH)RH ⊗ 1K

)QK

))RK

= R∗H⊗NK(1H⊗NK ⊗ T )(1H⊗NK ⊗QH ⊗ 1K)(1K ⊗RH ⊗ 1K)(1K ⊗QK)RK

= R∗H⊗NK(1H⊗NK ⊗ T )(1H⊗NK ⊗QH ⊗ 1K)(1K ⊗ 1H⊗NH ⊗QK)RH⊗K

= R∗H⊗NK(1H⊗NK ⊗ T )(1H⊗NK ⊗QH ⊗QK)RH⊗K

= τ rH⊗NK(T (QH ⊗QK)

),

which proves the formula, by uniqueness of the element QH⊗NK.

Let vi ∈ MorM (Hi,H)N such that v∗i vi = 1 and∑n

i=1 viv∗i = 1. We prove

that QH =∑n

i=1 viQHiv∗i . We start by proving that RH =

∑ni=1(vi ⊗

vi)RHi . For every ξ, η ∈ H0, we have that

⟨n∑i=1

(vi ⊗ vi)RHi(1), ξ ⊗ η

⟩=

n∑i=1

〈RHi(1), v∗i ξ ⊗ vi∗η〉

=n∑i=1

〈v∗i η, v∗i ξ〉

= 〈η, ξ〉= 〈RH(1), ξ ⊗ η〉 .


Then we get that for every T ∈ MorM (H)N ,

τ lH(T ) =n∑

i,j=1

R∗Hi(v∗i ⊗ vi

∗)(T ⊗ 1H)(vj ⊗ vj)RHj

=n∑i=1

R∗Hi(v∗i Tvi ⊗ 1Hi

)RHi

=n∑i=1

τ lHi(v∗i Tvi)

=n∑i=1

τ rHi(v∗i TviQHi)

=n∑i=1

R∗Hi(1Hi

⊗ v∗i TviQHi)RHi

=n∑i=1

R∗Hi(1Hi

⊗ v∗i )(vi∗vi ⊗ TviQHiv

∗i )(1Hi

⊗ vi)RHi

= R∗H

(1H ⊗

( n∑i=1

TviQHiv∗i

))RH ,

which proves the formula, once again by uniqueness of QH. Note for lateruse that we have also proven that

τ lH(T ) =n∑i=1

τ lHi(v∗i Tvi), for all T ∈ MorM (H)N . (5.7)

Finally, we prove that QH = (QH)−1. We start by proving that (T ⊗1H)RH = (1H ⊗ T

∗)RH, for every T ∈ MorM (H)N . For all ξ, η ∈ H0,

〈(T ⊗ 1H)RH(1), ξ ⊗ η〉 = 〈RH(1), T ∗ξ ⊗ η〉= 〈η, T ∗ξ〉= 〈Tη, ξ〉= 〈RH(1), ξ ⊗ T (η)〉= 〈(1H ⊗ T

∗)RH(1), ξ ⊗ η〉 .


So, we obtain that for all T ∈ MorN (H)M

τ lH(T ) = R∗H(T ⊗ 1H)RH= R∗H(1H ⊗ T

∗)RH= τ rH(T ∗)

= τ rH(T ) .

This formula implies that for every T ∈ MorN (H)M ,

τ rH(T ) = τ lH(TQ−1H ) = τ rH(T (QH)−1) .

Moreover, since τ rH(T ) = τ lH(T ), we have that τ lH(T ) = τ rH(T (QH)−1))and we have proven the required formula. Note for later use that we alsohave proven that

τ lH(T ) = τ rH(T ), for all T ∈ MorN (H)M . (5.8)

The foregoing proposition allows us to easily define a self-adjoint dimen-sion function on the class of finite index bimodules. More precisely weobtain the following theorem.

Theorem 5.32. Let H be a finite index M -N -bimodule and K a finiteindex N -P -bimodule. We use the notations of the previous proposition.Let α ∈ R. We define

dimα(H) := τ lH(Q−αH )

and we obtain that

• dimα(H⊗N K) = dimα(H) dimα(K) ,

• dimα(H⊕K) = dimα(H) + dimα(K) ,

• dimα(H) = dim1−α(H) .

So in particular, dim1/2 is a self-adjoint dimension function. Moreover,we have that

dim0(H) = dim(HN ) and dim1(H) = dim(MH) .

If H is irreducible, we have that

dimα(H) = dim(MH)α dim(HN)1−α .


Remark 5.33. If H and K are two finite index M -N -bimodules, we havedimα(H) = dimα(K) if H ∼= K.

Proof of the theorem. Remark that(MorM (H)N

)⊗N

(MorN (K)P

)⊂ MorM (H⊗N K)P

and the restriction of τ lH⊗NK is τ lH ⊗ τ lK. Indeed, because of (5.6), wehave, for every S ∈ MorM (H)N and T ∈ MorN (K)P

τ lH⊗NK(S ⊗ T ) = R∗H(1H ⊗R∗K ⊗ 1H)(S ⊗ T ⊗ 1)(1H ⊗RK ⊗ 1H)RH= τ lH(S)τ lK(T ) .

The multiplicativity of dimα is now obvious since we proved in Proposition5.31 that QH⊗NK = QH ⊗QK.

Let H := H1 ⊕ H2. Take isometries v1, v2 such that v1v∗1 + v2v∗2 = 1.

Because of Proposition 5.31 we obtain that

dimα(H) = τ lH

((∑i

viQHiv∗i

)−α)= τ lH

(∑i

viQ−αHiv∗i)

=∑i,j

τHj

(v∗j viQ

−αHiv∗i vj

),by (5.7)

=∑i

τHj (Q−αHi

)

=∑i

dimα(Hi) .

We prove that dimα(H) = dim1−α(H).

dimα(H) = τ lH(Q−αH )

= τ lH(QHα),by Proposition 5.31

= τ rH(QαH),by (5.8)

= τ rH(QαH), since QH is positive

= τ lH(Qα−1H ), by definition of QH

= dim1−α(H) .


Furthermore, we have that

dim0(H) = τ lH(1)

= (Tr⊗τ)(1), see the proof of Proposition 5.31

= dim(HN),by Remark 4.4 .

Similarly, we have that dim1(H) = τ lH(Q−1H ) = τ rH(1) = dim(MH).

Suppose now that H is irreducible. Then, MorM (H)N = C1 so in partic-ular, there exists λ > 0 such that QH = λ1. Then, we have that

dim(MH)α dim(HN)1−α = dim1(H)α dim0(H)1−α

= λ−α(τ lH(1))α(τ lH(1))1−α

= τ lH(Q−αH )

= dimα(H) .

Remark 5.34. Let N ⊂M be a finite index inclusion of II1 factors anddenote by [M : N ]0 the minimal index. We refer to [18, 33] for furtherdetails concerning minimal index. Then, it can be proven that

dim1/2

(ML2(M)N

)=√

[M : N ]0 .

Chapter 6

Fusion algebras and

bimodule categories

After the previous chapters, we have gathered enough material to under-stand both finite index bimodules over a II1 factor M and their intertwin-ers at the same time in a general setting. We will see that these data giverise to a C∗-tensor category Bimod(M). We will first focus on the set ofequivalence classes of finite index bimodules which already has a very richalgebraic structure: we prove that it is a fusion algebra. We recall somedefinitions and basic properties of abstract fusion algebras but we will bemostly dealing with the fusion algebra FAlg(M) of finite index bimodulesover a II1 factor M . The main aim of this section is to show that thefusion algebra FAlg(M) and, a fortiori the category Bimod(M), are ex-tremely rich invariants of the factor M . Their computation is in generalvery difficult but once it can be performed, we obtain lots of structural in-formations concerning M . Indeed, we will see that knowledge of FAlg(M)allows us to compute, for example, the fundamental group F(M) and alsothe outer automorphism group Out(M), the finite index subfactors...

6.1 The fusion algebra of a finite index bimodule

Definition 6.1. A fusion algebra A is a free N-module N[G] equippedwith the following additional structure:

102 Chapter 6. Fusion algebras and bimodule categories

• an associative and distributive product operation, and a multiplica-tive unit element e ∈ G ,

• an additive, anti-multiplicative, involutive map x 7→ x , called con-jugation,

satisfying Frobenius reciprocity: defining the numbers m(x, y; z) ∈ N forx, y, z ∈ G through the formula

xy =∑z

m(x, y; z)z ,

one has m(x, y; z) = m(x, z; y) = m(z, y;x) for all x, y, z ∈ G.

The base G of the fusion algebra A is canonically determined: these areexactly the non-zero elements of A that cannot be expressed as the sumof two non-zero elements. The elements of G are called the irreducibleelements of the fusion algebra A and we sometimes write G = IrredA .

Notice that conjugation preserves irreducibility.

The intrinsic group grp(A) of the fusion algebra A consists of the irre-ducible elements x ∈ A such that xx = e . Equivalently, x ∈ A belongsto the intrinsic group if and only if xx is irreducible. It is easy to checkthat the intrinsic group of a fusion algebra is indeed a group. If x, y ∈ A ,

we sometimes say that x is included in y, if there exists a z such thaty = x+ z .

Definition 6.2. Let A be a fusion algebra. A dimension function onA is an additive and multiplicative unital map d : A → R+ such thatd(x) = d(x).

One can also consider dimension functions that are not invariant underconjugation, see for example the dimension function dimα in Theorem5.32 when α 6= 1/2. Dimension functions satisfying this extra propertyare sometimes called self-adjoint.

From now on, all the fusion algebras that we consider are equipped with adimension function. Note that whenever x ∈ A is non-zero, e is includedin xx and so d(x) ≥ 1. It then follows that x ∈ A belongs to the intrinsicgroup of A if and only if d(x) = 1 . Moreover, if x ∈ A is non-zero andnot in the intrinsic group, the same reasoning yields d(x) ≥

√2 .

6.1 The fusion algebra of a finite index bimodule 103

This definition of a fusion algebra is very close to the notion of hyper-groups, see for example [51]. Two examples of fusion algebras arise asfollows.

• Let Γ be a group and define A = N[Γ]. The function d : Rep(G) →R∗

+ such that d(s) = 1 for all s ∈ Γ is a dimension function.

• Let G be a compact group and define the fusion algebra Rep(G)as the set of equivalence classes of finite dimensional unitary repre-sentations of G. The operations on Rep(G) are of course given bydirect sum and tensor product of representations, while the dimen-sion function d is given by the ordinary Hilbert space dimension ofthe representation space.

We define FAlg(M) as the class of finite index M -M -bimodules mod-ulo unitary equivalence, equipped with direct sums and Connes’ tensorproduct. The conjugate of an element of FAlg(M) is the contragredientbimodule.

Addition in FAlg(M) is given by the direct sum of bimodules, while mul-tiplication in FAlg(M) is given by the Connes tensor product of M -M -bimodules. Lemma 3.15 and Lemma 3.16 prove that ⊗M is an associativeand distributive operation while 3.17 proves that the bimodule L2(M)is a left and right multiplicative unit. The conjugation map is givenby the contragredient bimodule, it is anti-multiplicative by Proposition3.20. Finally, Frobenius reciprocity was proven in Proposition 5.30. Weconstructed a dimension function on FAlg(M) in Theorem 5.32.

6.1.1 Quasi-normalizers and bimodules

Let N ⊂M be an inclusion of II1 factors. The quasi-normalizer QNM (N)of N inside M is a unital ∗-subalgebra of M containing N that we definedin section 2.1. Quasi-normalizing elements can be characterized usingfinite index bimodules, as follows.

Lemma 6.3. Let N ⊂ M be an irreducible inclusion of II1 factors anda ∈M . The following are equivalent.

• a ∈ QNM (N).


• The bimodule NaN ⊂ L2(M) has finite index.

Proof. By definition of a quasi-normalizing element, we immediately havethat NaN is a finite index bimodule. Conversely, let a ∈ M and sup-pose that NaN has finite index. By Lemma 5.17 and its right-handedanalogue, we find vi, i = 1, . . . , n and wj , j = 1, . . . ,m in M such that

spanviN | i = 1, . . . , n = NaN = spanNwj | j = 1, . . . ,m .

It follows that a ∈ QNM (N).

Lemma 6.4. Let N ⊂ M be an irreducible, quasi-regular inclusion ofII1 factors. Then, NL2(M)N is the orthogonal direct sum of a family ofirreducible, finite index N -N -subbimodules Ki ⊂ L2(M), i ∈ I. WritingK0i := Ki ∩M , we have, for any choice of decomposition, spanK0

i | i ∈I = QNM (N).

Proof. Whenever a ∈ QNM (N), the closure of NaN inside L2(M) is anN -N -subbimodule of finite index. So, the linear span of all finite indexN -N -subbimodules of L2(M), is dense in L2(M). Hence, L2(M) can bedecomposed into an orthogonal direct sum of a family of irreducible, finiteindex N -N -subbimodules Ki ⊂ L2(M), i ∈ I. Write K0

i := Ki ∩M . ByLemma 5.17 and its right-handed analogue, we find vij , j = 1, . . . , ni andwij , j = 1, . . . ,mi in K0

i such that

spanvijN | j = 1, . . . , ni = K0i = spanNwij | j = 1, . . . ,mi .

Since NK0iN = K0

i , it follows that K0i ⊂ QNM (N).

The proof of Lemma 5.17 yields vi ∈ (Cni)∗⊗M such that the orthogonalprojection Pi of L2(M) onto Ki satisfies Pi(a) = vi(id⊗ EN )(v∗i a) for alla ∈M . In particular, Pi(M) = K0

i .

Let now a ∈ QNM (N) and decompose the closure of NaN inside L2(M)as a direct sum of irreducible N -N -subbimodules H1, . . . ,Hn. Lemma5.16 implies that for every i = 1, . . . , n, there exists a finite subset Ii ⊂ I

such that Hi 6∼= Kj whenever j ∈ I \ Ii. So, we find a finite subset I0 ⊂ I

such that NaN ⊂ spanKi | i ∈ I0. But then,

a =∑i∈I0

Pi(a) ∈ spanK0i | i ∈ I .


6.1.2 Outer automorphism groups, fundamental groupsand fusion algebras

The following proposition shows that the outer automorphism group ofthe II1 factor M and the fundamental group of M are encoded by finiteindex bimodules. We recall the definition of the fundamental group of aII1 factor. Let M be a II1 factor. We define the amplification of M bythe positive real t as follows. If t < 1, take p a projection of trace t in Mand define M t := pMp. If t > 1, taking a large enough integer n, we canfind a projection p ∈ Mn(C) ⊗M with non-normalized trace equal to t.In that case, we define M t := p(Mn(C) ⊗M)p. The fundamental groupof M is defined as

F(M) := t ∈ R∗+ |M t ∼= M .

It can be proven that F(M) is a multiplicative subgroup of R∗+.

Here we use again the notation 5.2 where we write finite index bimodulesas H(ψ), for some finite index inclusion ψ.

Proposition 6.5. Let M be type II1 factor.

• Every M -M -bimodule H in the intrinsic group grp(M) is isomor-phic with the bimodule H(π), for some ∗-isomorphism π : M →pMnp.

• We have a short exact sequence given by

e→ Out(M) → grp(M) → F(M) → e ,

mapping α ∈ Aut(M) to H(α) ∈ grp(M) and mapping H(π) ∈grp(M) to Tr(p).

Proof. The first assertion implies immediately that the map sendingH(π)∈ grp(M) to Tr(p) is surjective, by definition of the fundamental group.The map α 7→ H(α) is a homomorphism, because of Corollary 5.9, andinjective by Proposition 5.15. So, we are left with proving the first state-ment of the proposition.

Take H ∈ grp(M). Proposition 5.1 yields a finite index inclusion π :M → pMnp such that H ∼= H(π), as M -M -bimodules. We prove that π


is surjective. In Proposition 5.1 we proved that

dim(MH(π)) = (Tr⊗τ)(p) and dim(H(π)M) =[pMnp : π(M)]

(Tr⊗τ)(p).

In particular, dim(MH(π)) dim(H(π)M) = [pMnp : π(M)]. We use nowthe dimension function dim1/2 constructed in Theorem 5.32 and obtainthat

1 = dim1/2(L2(M))

= dim1/2(H(π)⊗M H(π))

= dim1/2(H(π))2

= dim(MH(π)) dim(H(π)M)

= [pMnp : π(M)] .

So, π(M) = pMnp (see Proposition 4.9).

6.1.3 Outer actions of countable groups and fusion alge-bras

We recall that an action σ of a countable group Γ on the II1 factor M issaid to be outer if for all g 6= e the automorphism σg is not inner. Werefer to section 2.4 for a basic introduction to outer actions.

Proposition 6.6. Consider an outer action Γ y M of a countable groupΓ on a II1 factor M . The map

N[Γ] → FAlg(M) : g 7→ H(σg)

defines an embedding of fusion algebras.

Proof. Using Corollary 5.9, one gets the M -M -bimodule isomorphismH(σg) ⊗M H(σh) ∼= H(σgh). Moreover, if H(σg) ∼= H(σh), Proposition5.15 yields a unitary u ∈ M such that σg = (Adu) σh which impliesg = h by outerness of the action.

When Γ acts outerly on the II1 factor M , every finite dimensional repre-sentation of Γ gives a finite index M -M -bimodule, where M := N o Γ.


Indeed, let π : Γ → U(k) be a finite dimensional representation of Γ. De-fine the M -M -bimodule Hrep(π) as the bimodule M

((Ck)∗ ⊗ L2(M)

)θπ(M)

withθπ : M → Mk(C)⊗M : aug 7→ π(g)⊗ aug .

In the particular case where Γ is a finite group we obtain the followinginteresting result.

Proposition 6.7. Let Γ be a finite group acting outerly on the II1 factorN . Then the map

Rep(Γ) → FAlg(N o Γ) : π 7→ Hrep(π)

is an embedding of fusion algebras.

Proof. Let M := N oΓ. Because of Proposition 5.8, we have Hrep(π)⊗MHrep(ρ) ∼= H

((id ⊗ θπ)θρ

). And an immediate computation shows that

(id⊗ θπ)θρ = θπ⊗ρ. The injectivity of the map π 7→ Hrep(π) follows fromthe equality

MorM (Hrep(π),Hrep(η))M ∼= Mor(π, η) ,

where Mor(π, η) consists of the linear maps T : Hπ → Hη such thatTπ(g) = η(g)T , for all g ∈ Γ. We prove now this equality. Let π, η ∈Rep(Γ) and denote n := dim(π), m := dim(η). Let T ∈ MorM

(Hrep(π),

Hrep(η))M

. Corollary 5.13 yields an element vT ∈ Cn(Cm)∗ ⊗M suchthat

θπ(x)v = vθη(x), for all x ∈M . (6.1)

Then, in particular, v commutes with 1⊗N and thus vT ∈ Cn(Cm)∗⊗ 1,by outerness of the action Γ y N . So, we write vT = Ω(vT ) ⊗ 1, withΩ(vT ) ∈ Cn(Cm)∗. By (6.1), it follows that Ω(vT )∗ ∈ Mor(π, η). Since Tis the operator RvT of right multiplication by vT (see Corollary 5.13) itis now obvious that the maps

MorM (Hrep(π),Hrep(η))M → Mor(π, η) : T 7→ Ω(vT )∗

andMor(π, η) → MorM (Hrep(π),Hrep(η))M : Ω 7→ RΩ∗⊗1

are each other’s inverse.


Given an arbitrary II1 factor it is quite hopeless to try to compute itsfusion algebra. Indeed, this fusion algebra is usually too big to handle: itcontains for example the outer automorphism group of the factor, whichcan be huge. We have seen that if the factor is a the crossed-productNoΓof N by a countable group Γ, every finite dimensional representation ofΓ gives an element in FAlg(N o Γ). This profusion of bimodules makesthe fusion algebra difficult to analyze, one has to decrease the amount ofbimodules, by imposing good conditions on the factor, in order to have achance to compute the entire fusion algebra. For example, FAlg(N o Γ)contains considerably less bimodules if we take Γ to be a group withoutany non-trivial finite dimensional unitary representations. The groupSL(3,Q) has such property, see for example [59].

This absence of finite dimensional representations can be formulated moreabstractly, in a bimodule language. When N ⊂ M is a quasi-regular in-clusion, L2(M) is a direct sum of finite index N -N -bimodules (see Propo-sition 6.4 for a more precise statement). So we can consider the fusionsubalgebra F0 of FAlg(N) generated by all finite index N -N -bimodulesthat appear in L2(M). Then, we have the following.

Lemma 6.8. Let N be a II1 factor and Γ y N an action of the countablegroup Γ by outer automorphisms of N . Put M := N o Γ. Then, thefollowing are equivalent

1. Whenever MKM is an irreducible finite index M -M -bimodule con-taining a non-zero element of F0, defined above, as N -N -subbimo-dule, we have MKM ∼= ML2(M)M.

2. Γ has no non-trivial finite dimensional unitary representations.

Remark 6.9. We do not use this lemma in this thesis but the characteri-zation 1 will be used in Chapter 10. To prove the main theorem of Chapter10, we needed to define a concept of “absence of finite dimensional rep-resentations” in a more general context that crossed products, we dealwith quasi-regular and irreducible inclusions N ⊂ M0. This lemma wasour motivation to work in this more abstract setting and one can keep inmind that the first characterization in the foregoing lemma only meansthat Γ is a group with no non-trivial finite dimensional representations,when M0 = N o Γ.


Proof. If π : Γ → U(k) is a non-trivial irreducible finite dimensionalunitary representation, define K = Hrep(π). Then, MKM is an irreduciblefinite index M -M -bimodule and MKM 6∼= ML2(M)M. Nevertheless, wehave the N -N -subbimodule (Ck)∗ ⊗ L2(N) ⊂ K, which is a sum of kcopies of the trivial N -N -bimodule and hence, belongs to F0. So, thefirst assertion fails.

Conversely, suppose that Γ has no non-trivial finite dimensional unitaryrepresentations and denote the action by ρ : Γ y N . Observe that theirreducible elements in F0 are precisely the N -N -bimodules H(ρs), s ∈ Γ,defined on the Hilbert space L2(N) with bimodule action

a · ξ · b := aξρs(b) for all ξ ∈ L2(N), a, b ∈ N .

Let MKM be an irreducible finite index M -M -bimodule. Let H ⊂ K bean N -N -subbimodule such that NHN ∼= NH(ρs)N for a certain s ∈ Γ. Byirreducibility of K, it follows that the span of all ur · H · uk, r, k ∈ Γ, isdense in K. So, K decomposes as a direct sum of N -N -subbimodules,each of them being isomorphic to one of the H(ρr), r ∈ Γ. Multiplyingon the right by ur, it follows that every N -M -subbimodule of K containsthe trivial N -N -bimodule as an N -N -subbimodule.

Because of Proposition 5.1, we can write MKM ∼= ψ(M)p(Cn ⊗ L2(M))M.Since K has finite index andN ⊂M is irreducible, the relative commutantA := ψ(N)′∩pMnp is finite dimensional; see Lemma 4.12. By the previousparagraph and Corollary 5.13, we find for every minimal projection q ∈A, a non-zero vector ξ ∈ q(Cn ⊗ L2(M)) satisfying ψ(a)ξ = ξa for alla ∈ N . As an element of q(Mn(C)⊗L1(M))q, the operator ξξ∗ commuteswith ψ(N) and hence, is a multiple of q; see section 2.5. So, we mayassume that ξ ∈ q(Cn ⊗M) and ξξ∗ = q. Since N ⊂ M is irreducible,ξ∗ξ = 1. Repeating this procedure for a family of minimal projectionsin A summing to 1, we find a unitary X ∈ q(Cn(Cm)∗ ⊗M) such thatX∗ψ(a)X = 1⊗ a for all a ∈ N .

So, we may assume that p = 1 and ψ(a) = 1⊗ a for all a ∈ N . But then,ψ(ug)(1⊗ u∗g) commutes with 1⊗N :

ψ(ug)(1⊗ u∗g)(1⊗ a) = ψ(ug)(1⊗ ρg−1(a))(1⊗ u∗g)

= ψ(ugρg−1(a))(1⊗ u∗g)

= (1⊗ a)ψ(ug)(1⊗ u∗g) .


By outerness of the action Γ y N , we write ψ(ug) = π(g) ⊗ ug for allg ∈ Γ and it is easily verified that π is a finite dimensional representationof Γ. Since Γ has no finite dimensional unitary representations, it followsthat π(g) = 1. By irreducibility of K, it follows that n = 1 and MKM ∼=ML2(M)M.

6.1.4 Fusion algebra of almost normalizing bimodules.

The notion of almost normalizing bimodules was introduced by Vaes in[54]. In [54], Vaes considers inclusions of II1 factors N ⊂ NoΓ, where Γ isa countable group acting outerly on N . He defines the fusion subalgebraof FAlg(N) consisting of finite index N -N -bimodules almost-normalizingΓ y N . All results gathered in this section come from [54] but wedefine here the notion of finite index bimodules almost-normalizing anirreducible and quasi-regular inclusion of II1 factors N ⊂ M . This gen-eralization is needed for Theorem 10.1.

Definition 6.10. Let N ⊂M be an irreducible and quasi-regular inclu-sion of type II1 factors. A finite index N -N -bimodule is said to almost-normalize the inclusion N ⊂ M , inside FAlg(N), if it arises as a finiteindex N -N -subbimodule of a finite index M -M -bimodule.

The following lemma clarifies the terminology of almost normalizing bi-modules and is obtained after Proposition 3.1 in [54].

Lemma 6.11. Let N be a II1 factor and σ : Γ y N an outer action.Let NHN be an N -N bimodule almost normalizing N ⊂ N o Γ insideFAlg(N). Then, there exists a finite index subgroup Λ ⊂ Γ such that forevery g ∈ Λ, there exist h, k ∈ Γ satisfying

H(σg)⊗N H ∼= H⊗N H(σh) and H⊗N H(σg) ∼= H(σk)⊗N H .

It is also shown in Lemma 4.1 in [54] that the fusion algebra generatedby the bimodules almost-normalizing a rigid inclusion N ⊂ N o Γ is acountable fusion subalgebra of FAlg(N). Following the lines of this proofwe get analogously

Lemma 6.12. Let N ⊂ M be a rigid, irreducible and quasi-regular in-clusion of type II1 factors. Then, the fusion algebra F generated by theN -N -bimodules almost-normalizing the inclusion N ⊂ M is a countablefusion subalgebra of FAlg(N).


In the proof we play the separability of M against the rigidity of theinclusion N ⊂ M to deduce the countability of F . This argument isfairly standard and is based on the following pigeon-hole trick.

Lemma 6.13. Let M be a separable von Neumann algebra (i.e L2(M) isseparable) and N a von Neumann algebra. Let

πi : N →M | i ∈ I

be an uncountable family of unital, normal ∗-homomorphisms. Let F ⊂ N

be a finite subset and ε > 0. Then there exists i 6= j in I such that for allx ∈ F

‖πi(x)− πj(x)‖2 < ε .

Proof. Let ε > 0. Separability of L2(M) yields a countable family ofballs Bn of diameter smaller than ε such that L2(M) =

⋃n∈NBn. Let

F := x1, . . . , xr ⊂ M be a finite subset of M and πi : N → M | i ∈I be an uncountable family of unital, normal ∗-homomorphisms. Thepigeon-hole principle yields an uncountable subset I1 ⊂ I and an integern1 ∈ N such that πi(x1) ∈ Bn1 , for all i ∈ I1. Uncountability of I1 yields,once again by the pigeon-hole principle, an uncountable subset I2 ⊂ I1and an integer n2 such that πi(x2) ∈ Bn2 , for all i ∈ I2. Then, we obtaininductively a family of uncountable subsets I ⊃ I1 ⊃ I2 ⊃ · · · ⊃ Ir andintegers n1, . . . , nr ∈ N such that

πj(xi) ∈ Bni , for all j ∈ Ii, and all i = 1, . . . , r .

Let i 6= j in Ir. Then we have that πi(xk) ∈ Bnkand πj(xk) ∈ Bnk

, forall k ≤ r, since Ik ⊃ Ir. The conclusion follows.

Proof of Lemma 6.12. LetH be an irreducible finite indexM -M -bimodu-le containing a finite index N -N -subbimodule K. By quasi-regularity ofN ⊂M , it follows that the linear span of all a ·K · b, with a, b ∈ QNM (N)is dense in H. So H decomposes as a direct sum of finite index N -N -bimodules and hence, every finite index M -M -bimodule containing afinite index N -N -subbimodule decomposes as a direct sum of finite indexN -N -subbimodules.

Suppose that F is uncountable. So, there exist uncountably many finiteindex M -M -bimodules, disjoint as N -N -bimodules, that we write

Hj∼= ψj(M)

(pj(Cnj ⊗ L2(M)

))M, for j ∈ J .


Denote by Tr the non-normalized trace on Mnj (C). Since nj can be chosenas large as we want, we may also assume that (nj)−1 ≤ (Tr⊗τ)(pj) aswell. Since (nj)j∈J is an uncountable collection of integers, the pigeonhole principle yields an uncountable subset I ⊂ J and an integer n ∈ Nsuch that ni | i ∈ I is bounded by n. So we have an uncountable familyof M -M -bimodules (Hi), disjoint as N -N -bimodules, satisfying, for alli ∈ I,

Hi∼= ψi(M)

(pi(Cn ⊗ L2(M)

))M, with

1n≤ (Tr⊗τ)(pi)

and containing the finite index irreducible N -N -subbimodule Ki. Wereach a contradiction by proving the existence of i 6= j such that Ki ∼= Kj .

Because of Lemma 2.14, rigidity of the inclusion N ⊂ M yields an ε > 0and a finite subset F ⊂M such that every M -M -bimodule H that admitsa vector ξ ∈ H, with 1− ε ≤ ‖ξ‖2 ≤ 1, and satisfying

|〈ξ, a · ξ · b〉 − τ(ab)| < ε, for all a, b ∈ F , (6.2)

has a non-zero N -central vector ξ0.

We may assume that 1 ∈ F and view the ψi : M →Mn as non-unital ∗-homomorphisms such that ψi(1) = pi. We may also suppose that F = F ∗

and F ⊂ (M)1. The above pigeon-hole Lemma 6.13 yields elements i 6= j

such that ‖ψi(x)− ψj(x)‖2 <εn , for all x ∈ F . So we have

‖ψi(x)− ψj(x)‖2 ≤ε√n

(Tr⊗τ)(pi)1/2 ≤ ε‖pi‖2 .

We consider theM -M -bimodule pi(Mn(C)⊗L2(M)

)pj with left and right

actions given by

a · η · b := ψi(a) η ψj(b), for all a, b ∈ M .

We prove that the vector ξ := ‖pi‖−12 pipj satisfies (6.2). By uniqueness

of the trace on M , we have that (Tr⊗τ) ψi = (Tr⊗τ)(pi)τ = n‖pi‖22τ .


So, we obtain, for all a, b ∈ F ,

〈ξ, a · ξ · b〉 − τ(ab)

= ‖pi‖−22 〈pipj , ψi(a)ψj(b)〉 − τ(ab)

= ‖pi‖−22 (tr⊗τ)

(ψi(a)ψj(b)

)− τ(ab)

= ‖pi‖−22 (tr⊗τ)

(ψi(a)ψj(b)

)− ‖pi‖−2

2 (tr⊗τ)(ψi(ab)

)= ‖pi‖−2

2 (tr⊗τ)(ψi(a)(ψj(b)− ψi(b)

)≤ ‖pi‖−2

2 ‖ψi(a∗)‖2‖ψj(b)− ψi(b)‖2

= ‖pi‖−12 ‖a‖2‖ψj(b)− ψi(b)‖2

≤ ‖a‖2ε

≤ ε .

The rigidity of the inclusion N ⊂M yields a non-zero N -central vector

v ∈ ψi(M)

(pi(Mn(C)⊗ L2(M)

)pj

)ψj(M) .

Since the inclusion ψi(M) ⊂ piMnpi has finite index and N ⊂ M is

irreducible, the algebra Ai := ψi(N)′ ∩ piMnpi is finite dimensional;

see Lemma 4.12. As an element of pi(Mn(C) ⊗ L1(M)

)pi, the opera-

tor vv∗ belongs to Ai and is then bounded. So, we may assume thatv ∈ pi

(Mn(C) ⊗M

)pj . Then, the operator of left multiplication by v

yields a non trivial (since v is non-trivial) intertwiner

T : ψj(N)

(pj(Cn ⊗ L2(M)

))N → ψi(N)

(pi(Cn ⊗ L2(M)

))N .

So, ψj(N)

(pj(Cn ⊗ L2(M)

))N and ψi(N)

(pi(Cn ⊗ L2(M)

))N are not dis-

joint N -N -bimodules. We have reached a contradiction.

6.1.5 Freeness and free products of fusion algebras

The notions of freeness and free product of fusion algebras were intro-duced in [2, Section 1.2], in the study of free composition of subfactors.

Definition 6.14 ([2, Section 1.2]). Let A be a fusion algebra and Ai ⊂ Afusion subalgebras for i = 1, 2. We say that A1 and A2 are free inside Aif every alternating product of irreducibles in Ai\e, remains irreducibleand different from e.


Given fusion algebras A1 and A2, there is up to isomorphism a uniquefusion algebra A generated by copies of A1 and A2 that are free. We callthis unique A the free product of A1 and A2 and denote it by A1 ∗ A2.

Of course, the free product can be constructed in a concrete way as fol-lows: given A1 and A2, set Gi = Irred(Ai). Define G as the set of wordswith letters alternatingly from G1 \ e and G2 \ e. Denote the emptyword as e. Then, A1 ∗ A2 = N[G]. The product on N[G] is the uniqueassociative and distributive operation satisfying the following two condi-tions:

• The embeddings Ai → N[G] are multiplicative.

• If the last letter of the alternating word x ∈ G and the first letter ofthe alternating word y ∈ G belong to different fusion algebras Ai,the product of x and y is again irreducible and given by concatena-tion of x and y.

Denote by R the hyperfinite II1 factor. The fusion algebra FAlg(R) ishuge, in the sense that FAlg(R) contains many free fusion subalgebras.More precisely, it was shown in Theorem 5.1 of [54] that countable fusionsubalgebras of FAlg(R) can be made free by conjugating one of them withan automorphism of R (see Theorem 6.15 below). Note that the sameresult has first been proven for countable subgroups of Out(R) in [21]. Inboth cases, the key ingredients come from [38].

Let M be a II1 factor and MKM ∈ FAlg(M). Whenever α ∈ Aut(M), wedefine the conjugation of K by α as the bimodule

Kα := H(α−1)⊗M K ⊗M H(α) .

Theorem 6.15 (Theorem 5.1 in [54]). Let R be the hyperfinite II1 factorand A0,A1 two countable fusion subalgebras of FAlg(R). Then,

α ∈ Aut(R) | Aα0 and A1 are free

is a Gδ-dense subset of Aut(R).

Remark 6.16. Because of Proposition 5.8, the bimodule Kα has K asits underlying Hilbert space with new left and right module action givenby

ξ ·new a = ξ ·old α(a) and a ·new ξ = α(a) ·old ξ .

6.2 C∗-tensor categories and finite index bimodules 115

Theorem 6.15 is a key ingredient for the work presented in Chapter 9 andChapter 10. In both cases, our main theorem is based on the assumptionthat two countable fusion subalgebra F1 and F2 of the fusion algebraFAlg(R) of the hyperfinite II1 factor are free. Then, Theorem 6.15 tels usthat the set of automorphisms α of R such that Fα

1 and F2 are free is a Gδdense. So, in particular, non-empty. Our main results are the Theorem9.1 and Theorem 10.7 and they are both existence results since the proofuses Theorem 6.15, which is based on a Baire Category argument.

So, countable fusion subalgebras of FAlg(R) are very interesting to us.We have already met several examples.

• Let Γ y R outerly. Then, N[Γ] embeds into FAlg(R), by Proposi-tion 6.6.

• Let G y R minimally. Then Rep(G) embeds into FAlg(R), byTheorem 7.24.

• Let F be the fusion subalgebra of FAlg(R) consisting of bimodulesalmost normalizing a rigid, irreducible and quasi-regular inclusionR ⊂M , by Lemma 6.12.

We end this section with a last example. The following proposition is aneasy version of a more general result of Popa, see [36, Theorem 4.4].

Proposition 6.17. Let Γ y R be an outer action of the countable groupΓ on the hyperfinite II1 factor R. Denote M := R o Γ and assume thatthe inclusion R ⊂M is rigid. Then,

G :=Aut(R ⊂ Ro Γ)

U(R)

is a countable group.

Then, by restricting elements of G to R, we view G as a subgroup ofOut(R) which is itself embedded into FAlg(R), by Proposition 6.5.

6.2 C∗-tensor categories and finite index bimod-

ules

In this section, we do not aim to go into the theory of abstract C∗-tensorcategories. We remark that the bimodule category Bimod(M) consisting


of all finite indexM -M -bimodules over the II1 factorM fits in this generalsetting. We proved in section 5.6 that every finite index bimodules admitsa pair of conjugates (RH, RH). In the terminology of [12], Bimod(M) isa monoidal C∗-category with conjugates. We refer to [12] for furtherdetails concerning these categories, but following for example [26], weuse the terminology of C∗-tensor categories with conjugates. We alsorefer to [27] for terminology and definitions of category theory. Anotherexample of C∗-tensor category with conjugates is the category Rep(G)of all finite dimensional representations of the compact group G. In thisthesis, we mainly focus on this example and its relation with the categoryof bimodules over a II1 factor. In order to be able to compare C∗-tensorcategories with conjugates, we recall that a tensor functor between twosuch categories C and D is a functor

F : C → D ,

with the following additional properties.

• F (a⊗ b) and F (a)⊗ F (b) are naturally isomorphic ,

• F : Mor(a, b) → Mor(F (a), F (b)) is a linear map satisfying F (T ∗) =F (T )∗ ,

• F (a) and F (a) are naturally isomorphic .

A tensor functor F between C∗-tensor categories is called fully faithfulif it induces an isomorphism between Mor(a, b) and Mor(F (a), F (b)), forevery objects a and b.

A fully faithful tensor functor F : C → D between C∗-tensor categorieswith conjugates with the property that for every object b in D, there existsan object a in C such that F (a) is isomorphic to b is an equivalence ofcategories. Concerning equivalence of categories, we refer to [27, ChapterIV].

Minimal actions of compact groups and the category of bi-modules

Minimal actions of compact groups are studied in Chapter 7, we onlygive the definition. A strongly continuous action G y M of a compact

6.2 C∗-tensor categories and finite index bimodules 117

group G on a II1 factor M is said to be minimal if the map G→ Aut(M)is injective and if M ∩ (MG)′ = C1. Here, MG is the von Neumannalgebra of G-fixed points in M . We denote by Hπ the Hilbert space ofthe representation π.

We state here the main Theorem proven in Chapter 7. This theoremgoes back to [47] in the study of algebraic quantum field theory and canbe stated explicitly as follows.

Theorem 6.18 (Theorem 7.24). Let G be a second countable compactgroup and σ : G y M a minimal action. Set P := MG. Then,

Rep(G) → Bimod(P ) : π 7→ PMor(L2(M),Hπ)P , where

〈S, T 〉 := τ(S∗T ) and (a · S · b)(ξ) = S(a∗ξb∗) ,

for all S, T ∈ Mor(L2(M),Hπ), a, b ∈ P , ξ ∈ L2(M), defines a fullyfaithful tensor functor from the category Rep(G) of finite dimensionalunitary representation of G to the category Bimod(P ) of finite index P -P -bimodules.

In Chapter 10 we prove the existence of factors of type II1 M and of aminimal action G y M such that the functor Rep(G) → Bimod(P ) is anequivalence of C∗-tensor categories.

Chapter 7

Minimal actions of compact

groups

7.1 Representation theory for compact groups:

some basics notions

This first section gathers basic facts concerning the representation theoryof compact groups and should not be seen as an introduction to thistheory, we refer to [19] for the necessary background material.

Conventions. For our purpose, we always assume that our compactgroups are second countable. We consider representations π : G→ U(Hπ)of compact groups G on Hilbert spaces Hπ, that are strongly continu-ous, in the sense that the map g 7→ π(g)ξ is continuous for all ξ ∈ Hπ.We always denote Hπ the Hilbert space of the representation π and ε

denotes the trivial representation. We only consider unitary representa-tions because every representation of a compact group can be unitarized,considering the following scalar product:

〈ξ, η〉′ :=∫G〈π(g)ξ, π(g)η〉 dg ,

where dg is the Haar measure on G.

Intertwiners. Given two representations π and ρ, we denote by Mor(π,ρ) the space of operators T : Hπ → Hρ satisfying

Tπ(g) = ρ(g)T, for all g ∈ G .

120 Chapter 7. Minimal actions of compact groups

We recall Schur’s Lemma. Let π and ρ be irreducible representations ofG and T ∈ Mor(π, ρ). Then, either T = 0, either T is an isomorphism.

The set of irreducible representations is denoted by Irr(G).

Two representations π and ρ are said to be equivalent, and we writeπ ∼ ρ, if Mor(π, ρ) contains a unitary operator. Note that every isomor-phism in Mor(π, ρ) entails the unitary equivalence of π and ρ, by polardecomposition.

Every representation of G decomposes as a direct sum of irreducible ones.Moreover, irreducible representations of G are finite dimensional.

Contragredient. The contragredient representation π associated to therepresentation π is defined by π(g) := π(g).

Peter-Weyl’s theorem. To every irreducible representation π one as-sociates its coefficients Cπξ,η defined by

Cπξ,η(g) := 〈π(g)ξ, η〉, for all g ∈ G , (7.1)

where ξ, η ∈ Hπ. Given an orthonormal basis (ei)1≤i≤dimπ of Hπ, wecall coordinate functions associated to π the functions Cπjk := Cπek,ej

. Wedefine the trigonometric polynomials on G as the elements of the space

T (G) := spanCπξ,η | π ∈ Irr(G), and ξ, η ∈ Hπ .

We prove now Peter-Weyl’s theorem.

Theorem 7.1. The space of trigonometric polynomials on G is uniformlydense in C(G), the continuous functions on G.

Proof. We prove that the elements of T (G) separate the points of G andthat T (G) is closed under conjugation. Gelfand-Raikov’s Theorem im-plies that, given g 6= h in G, there exists an irreducible representationπ such that π(g) 6= π(h). Then, there exist vectors ξ, η ∈ Hπ such that〈π(g)ξ, η〉 6= 〈π(h)ξ, η〉. The space T (G) is stable under conjugation be-cause 〈π(g)ξ, η〉 = 〈π(g−1)η, ξ〉. Because of Stone-Weierstrass theorem, itonly remains to prove that T (G) is an algebra. Let π and ρ be irreduciblerepresentations. We decompose the representation π ⊗ ρ in a direct sumof irreducible representations as

π ⊗ ρ =n⊕i=1

τi . (7.2)

7.1 Representation theory for compact groups: some basicsnotions 121

Let ξ, ξ′ ∈ Hπ and η, η′ ∈ Hρ. Then, by definition of the tensor productrepresentation:

〈π(g)ξ, ξ′〉〈ρ(g)η, η′〉 = 〈(π ⊗ ρ)(g)(ξ ⊗ η), ξ′ ⊗ η′〉 .

Using equation (7.2), we know that 〈(π ⊗ ρ)(g)(ξ ⊗ η), ξ′ ⊗ η′〉 is a linearcombination of coefficients of τi. This ends the proof.

Orthogonality relations. Consider two representations π and ρ and abounded linear map L : Hπ → Hρ. We define, for all ξ ∈ Hπ and η ∈ Hρ

the complex number:

CL(η, ξ) :=∫G〈η, ρ(g−1)Lπ(g)ξ〉 dg .

The continuity of π and ρ imply that the integrand is a continuous func-tion on G and hence, CL(η, ξ) ∈ C. For all η ∈ Hρ, the map ξ 7→ CL(η, ξ)is a bounded linear functional on Hπ. This yields the existence of abounded linear map AL : Hπ → Hρ such that CL(η, ξ) = 〈η,ALξ〉.

Lemma 7.2. Let π and ρ be two irreducible representations of G and abounded linear map L : Hπ → Hρ. Then,

AL =

1dimπ (TrL)idHπ if π = ρ and Hπ = Hρ

0 if π and ρ are not equivalent

where Tr denotes the non-normalized trace on B(Hπ).

Proof. We first prove that AL intertwines the representations π and ρ.

〈η,ALπ(g)ξ〉 =∫G〈η, ρ(h−1)Lπ(hg)ξ〉 dh

=∫G〈η, ρ(gh−1)Lπ(h)ξ〉 dh

= 〈η, ρ(g)ALξ〉

If AL is non-zero and π = ρ, the Schur lemma yields a complex numberα such that AL = α id. We fix (ei)1≤i≤dimπ an orthonormal basis of Hπ


and compute α.

Tr(L) =∫G

Tr(π(g−1)Lπ(g)) dg

=dimπ∑i=1

∫G〈ei, π(g−1)Lπ(g)ei〉 dg

=dimπ∑i=1

〈ei, ALei〉 = α dimπ .

The equalities obtain in the following theorem are called orthogonalityrelations.

Theorem 7.3. Let π be an irreducible representation of G and (ei) anorthonormal basis for Hπ. Then∫

G〈em, π(g)ej〉〈ek, π(g)el〉 dg =

1dimπ

δj,lδk,m .

Proof. Denote by (eij) the matrix units in B(Hπ). Let k,m ∈ 1, . . . d.Then, using Lemma 7.2, we find that Aemk

= 1dimπ δk,mid. Easy compu-

tations lead to the equality

〈π(g)ej , emkπ(g)el〉 = 〈em, π(g)ej〉〈ek, π(g)el〉 .

The theorem follows by integrating both sides over G and using the factthat Aemk

= 1dimπ δk,mid.

7.2 Spectral subspaces

In this section we consider continuous actions, not necessarily minimal, ofcompact groups on von Neumann algebras. We investigate some generalproperties of the spectral subspace associated to a finite dimensional rep-resentation of a compact group. These subspaces are important tools tounderstand the action. Spectral subspaces are studied more intensivelyin section 7.3, in the context of minimal actions. They are the key in-gredient to build the faithful tensor functor Rep(G) → Bimod(MG) inTheorem 7.24 and one of the main results of the present section is thatspectral subspaces are finite index MG-MG-bimodules.

7.2 Spectral subspaces 123

7.2.1 General facts

Definition 7.4. Let G be a compact group and (M, τ) a tracial vonNeumann algebra. We call action of G on M a strongly continuous ho-momorphism σ : G→ Aut(M).

Convention. In this section σ : G y M always denotes a continuousaction σ of the compact group G on the tracial von Neumann algebra(M, τ). We denote by Mσ the fixed point algebra. When no confusionwith another action is possible, we only write MG := Mσ.

Definition 7.5. Let π be a finite dimensional representation of G andσ : G y M a continuous action. We denote by

• Mor(Hπ,M), the space of linear maps S : Hπ → M satisfyingσg S = S π(g), for all g ∈ G.

• L0(π) ⊂M , the linear span of Mor(Hπ,M)Hπ.

• L(π), the closure of L0(π) inside L2(M).

The space L(π) is called the spectral subspace of π.

Proposition 7.6. Let G y M be a continuous action. Then

spanL0(π) | π ∈ Rep(G)

is an ultraweakly dense ∗-algebra in M .

Proof. For every element S ∈ Mor(Hπ,M), we define QS ∈ H∗π⊗M such

that QS(ξ ⊗ 1) = S(ξ), for every ξ ∈ Hπ. The map

Mor(Hπ,M) → H∗π ⊗M : S 7→ QS

yields an isomorphism

L0(π) ∼= ξ(Hπ⊗1) | ξ ∈ H∗π⊗M and (id⊗σg)(ξ) = ξ(π(g)⊗1) . (7.3)

We prove now that spanL0(π) | π ∈ Rep(G) is unital a ∗-algebra. Letπ, η ∈ Rep(G) and take ξ ∈ H∗

π⊗M , µ ∈ H∗η⊗M satisfying (id⊗σg)(ξ) =

ξ(π(g)⊗ 1) and (id⊗ σg)(µ) = µ(η(g)⊗ 1). Then, we have(ξ(Hπ ⊗ 1)

)(µ(Hη ⊗ 1)

)= ξ13µ23(Hπ⊗η ⊗ 1)


and (id⊗ id⊗ σg

)(ξ13µ23

)= ξ13µ23

(π(g)⊗ η(g)⊗ 1

),

proving that spanL0(π) | π ∈ Rep(G) is an algebra, by (7.3).

We prove now that L0(π)∗ = L0(π), for every π ∈ Rep(G). Let J be thecanonical anti-unitary on L2(M). We also denote by

J : Hπ → Hπ : ξ → ξ .

By definition, the map

θ : Mor(Hπ,L2(M)) → Mor(Hπ,L2(M)) : T 7→ JTJ (7.4)

is an anti-linear isomorphism. We have

L0(π)∗ = J(Mor(Hπ,M)Hπ

)= J

(Mor(Hπ,M)

)J (Hπ)

= θ(Mor(Hπ,M)

)Hπ

∼= L0(π) .

We have proven that spanL0(π) | π ∈ Rep(G) is a unital ∗-algebra soit is enough to prove that it is L2-dense in M to obtain the proposition.Let π be a finite dimensional representation of G. We define

X := span∫

G〈π(g)ξ, µ〉σg(x) dg | π ∈ Rep(G), ξ, µ ∈ Hπ, x ∈M

.

Let ξ, µ ∈ Hπ and x ∈M . Because of the invariance of the Haar measureand formula (7.3), the element∫

G〈π(g)ξ, µ〉σg(x) dg =

(∫Gξ∗π(g)∗ ⊗ σg(x) dg

)(µ⊗ 1)

belongs to L0(π). This proves that X ⊂ span(⋃

π∈Rep(G) L0(π)

). We

prove now that the subspace X is L2-dense in M . Since the linear spanof

g 7→ 〈π(g)ξ, µ〉 | π ∈ Irr(G), ξ, µ ∈ Hπ

is dense in C(G) (see Theorem 7.1), it is enough to prove that the set

span∫

Gf(g)σg(x) dg | x ∈M, f ∈ C(G)


is dense in M .

Take ε > 0 and x ∈ M . There exists a neighborhood V of the identityelement e ∈ G such that

‖x− σg(x)‖2 < ε, for all g ∈ V .

Now, we take f ∈ C(G). We may assume f is a positive function sup-ported on V such that

∫G f = 1. Then:∥∥∥∥x− ∫

Gf(g)σg(x) dg

∥∥∥∥2

=∥∥∥∥∫

Gf(g)(x− σg(x)) dg

∥∥∥∥2

< ε .

7.2.2 MG-MG-bimodules

Definition-Proposition 7.7. Let σ : G y (M, τ) be a continuous actionand denote P := MG, the fixed-point algebra. We extend the actionG y M to a unitary representation σ : G→ U(L2(M)). Let π be a finitedimensional representation of G. Then, the space

Mor(Hπ,L2(M)) :=T : Hπ → L2(M) | σg(T (ξ)) = T (π(g)ξ),

for all ξ ∈ Hπ, g ∈ G ,

endowed with the scalar product 〈S, T 〉 := Tr(S∗T ) is a P -P -bimodule,for the following left and right P -actions:

(a · S · b)(ξ) := aS(ξ)b, for all S ∈ Mor(Hπ,L2(M)), a, b ∈ P, ξ ∈ Hπ .

Similarly, the space

Mor(L2(M),Hπ) :=T : L2(M) → Hπ | T (σg(ξ)) = π(g)T (ξ),

for all ξ ∈ L2(M), g ∈ G

endowed with the scalar product 〈S, T 〉 := τ(S∗T ) is a P -P -bimodule, forthe following left and right P -actions:

(a·S·b)(ξ) := S(a∗ξb∗), for all S ∈ Mor(L2(M),Hπ), a, b ∈ P, ξ ∈ L2(M).


Remark that the map S 7→ S∗ is an isomorphism of P -P -bimodules

PMor(Hπ,L2(M))P ∼= PMor(L2(M),Hπ)P .

Note that the the inner product 〈S, T 〉 of two elements in Mor(Hπ,L2(M))can be computed using the formula

〈S, T 〉 =dimπ∑i=1

〈S(ei), T (ei)〉 , (7.5)

for any orthonormal basis of (ei) of Hπ. This number is independent ofthe choice of the orthonormal basis. Indeed, if (fi) is another orthonormalbasis of Hπ then take a unitary u ∈ U(Hπ) such that u(ei) = fi, we havethat

∑i〈S(fi), T (fi)〉 = Tr(u∗S∗Tu) = 〈S, T 〉.

Lemma 7.8. Let G y (M, τ) be a continuous action. Then,

Mor(Hπ,L2(M))0 = Mor(Hπ,M) .

As a consequence Mor(Hπ,M) is dense in Mor(Hπ,L2(M)). Moreover,if we denote by (e1, . . . , edimπ) an orthonormal basis of Hπ, we have that

〈S, T 〉P =dimπ∑i=1

S(ei)∗T (ei) ,

and this does not depend on the choice of the orthonormal basis.

Proof. Let (e1, . . . , edimπ) an orthonormal basis of Hπ. We start by prov-ing that

∑i S(ei)∗T (ei) ∈ P , for every S, T ∈ PMor(Hπ,M)P . We first

prove that∑

i S(ei)∗T (ei) is independent of the choice of the basis (ei).Let (fi) be another orthonormal basis of Hπ and u the unitary satisfyingu(ei) = fi. Then,∑

j

S(fj)∗T (fj) =∑i,j,k

uijukjS(ei)∗T (ek)

=∑i,k

∑j

ukj(u∗)ji

S(ei)∗T (ek)

=∑i

S(ei)∗T (ei) .


Let g ∈ G. Take now the orthonormal basis defined by (fi) := (π(g)ei).We obtain

σg

(∑i

S(ei)∗T (ei)

)=∑i

S(fi)∗T (fi) =∑i

S(ei)∗T (ei) .

Take now an element T ∈ Mor(Hπ,L2(M))0. Reasoning as in the previousparagraph yields that

η :=∑i

T (ei)T (ei)∗ ∈ L1(P )+ .

There exists a constant c > 0 such that

‖a · T‖2 ≤ c‖a‖22, for all a ∈ P . (7.6)

We also have that

‖a · T‖2 =∑i

‖aT (ei)‖22

=∑i

τ(aT (ei)T (ei)∗a∗)

= τ(aηa∗) .

Because of (7.6), it follows that

‖η1/2a∗‖22 = τ(aηa∗)

≤ c‖a‖22 ,

so η1/2 is bounded and thus η ∈ P . Then, T (ei)T (ei)∗ is also bounded,since it is smaller than η. By polar decomposition, all the T (ei) belongto M and the lemma is proven.

We prove now that L(π) is a finite index MG-MG-bimodule.

Proposition 7.9. Let σ : G y (M, τ) be a continuous action. Then,Mor(Hπ,L2(M)) is a finite index MG-MG-bimodule with

dim(MGMor(Hπ,L2(M))

)≤ dim(π) and

dim(Mor(Hπ,L2(M))MG

)≤ dim(π) .

As a consequence, L(π) is a finite index MG-MG-bimodule satisfying

dim(MGL(π)) ≤ dim(π)2 and dim(L(π)MG) ≤ dim(π)2 .


We will need the following little lemma to prove Proposition 7.9.

Lemma 7.10. Let (P, τ) be a von Neumann algebra with the faithfulnormal trace τ . Let p, q be projections in P . If we equip pPp with thetracial state τ(·)

τ(p) , then

dim((q L2(P )p)pPp

)=τ(qzp)τ(p)

,

where zp denotes the central support of p in P .

Note that since P is not necessarily factorial, the dimension computed inthe foregoing lemma depends on the trace τ that P is endowed with, seeRemark 4.11. In our case this is not a problem since we work with M ,endowed from the beginning with the trace τ .

Proof of Lemma 7.10. Take partial isometries vi ∈ P with qi := v∗i vi ≤ p

and∑

i viv∗i = qzp. Then

q L2(P )p = qzp L2(P )p =⊕i

viv∗i L2(P )p .

Sinceviv

∗i L2(P )p→ qi L2(pPp)p : x 7→ v∗i x

is an isomorphism of right pPp-modules, we have that

dim((q L2(P )p)pPp

)=∑i

τ(qi)τ(p)

=τ(qzp)τ(p)

.

Proof of Proposition 7.9. Let σ : G y (M, τ) and π ∈ Rep(G). We define

M :=(

M H∗π ⊗M

Hπ ⊗M B(Hπ)⊗M

)with the action β : G y M defined by

βg

(a ξ

η b

)=(

σg(a) (id⊗ σg)(ξ)(π(g)∗ ⊗ 1)(π(g)⊗ 1)(id⊗ σg)(η) (Adπ(g)⊗ σg)(b)

).


The von Neumann algebra M is equipped with the faithful trace τ givenby

τ

(a ξ

η b

)=τ(a) + (Tr⊗τ)(b)

1 + dim(π),

where Tr denotes the non-normalized trace on B(Hπ). Let P := Mβ and

p :=(

1 00 0

)and q :=

(0 00 1

).

Then, MG = pPp and, by (7.3), we have that

MGMor(Hπ,L2(M)) ∼= pPp(pL2(P )q)

as left pPp-modules. Because of Lemma 7.10 we obtain

dim(MG

(Mor(Hπ,L2(M))

))≤ τ(q)τ(p)

=(

dim(π)1 + dim(π)

)(1 + dim(π))

= dim(π) .

In (7.4) we defined the anti-linear isomorphism

θ : Mor(Hπ,L2(M)) → Mor(Hπ,L2(M)) : T 7→ JTJ ,

where J is the canonical anti-unitary on L2(M) and J is the contragredi-ent map on Hπ. It follows straightforwardly from the definition of θ thatθ(T · a) = a∗ · θ(T ), for every a ∈ MG. So θ is an anti-isomorphism ofMG-MG-bimodules. Hence,

dim(Mor(Hπ,L2(M))MG

)= dim

(MGMor(Hπ,L2(M))

)≤ dim(π)

= dim(π) .

Moreover, by definition of L(π), the map

Hπ ⊗Mor(Hπ,L2(M)) → L(π) : ξ ⊗ T 7→ T (ξ)

is an isomorphism and hence

dim(MGL(π)) ≤ dim(π)2 and dim(L(π)MG) ≤ dim(π)2 .


7.3 Minimal actions and bimodule categories

Definition 7.11. A strongly continuous action G y M of a compactgroup G on a II1 factor M is said to be minimal if

• the map G→ Aut(M) is injective and,

• if M ∩ (MG)′ = C1.

Here, MG is the von Neumann algebra of G-fixed points in M .

Every second countable compact group G admits a minimal action on thehyperfinite II1 factor. Before giving a possible construction, we recall twoclassical results.

1. Let Λ be an ICC group acting outerly on the II1 factor N . Then,L(Λ)′ ∩N o Λ = C1.

2. Let ρ1 : Λ y N be an outer action of the countable group Λ on theII1 factor N and ρ2 : G y N be a continuous action of the compactgroup G. We suppose that ρ1 and ρ2 commute. Then,

ρ3 : G y N o Λ : ρ3(g)(auλ) = ρ2(g)(a)uλ

defines a continuous action of G on N o Λ.

We use the foregoing results to construct a minimal action of the compactgroup G on the hyperfinite II1 factor R. Take an amenable ICC groupΛ. Let Λ act on

∏Λ(G,Haar) via Bernoulli shift. So we have an action

Λ y N := L∞(∏

Λ(G,Haar)). Let G act on

∏Λ(G,Haar) by diagonal

left translation and extend this action to G y N . It is easy to verifythat the actions Λ y N and G y N commute. So, by point 2 above, wehave an action G y R := N o Λ. By construction, this action leaves Λpointwise invariant so we have L(Λ) ⊂ RG. Then,

(RG)′ ∩R ⊂ L(Λ)′ ∩R = C1 ,

because of the point 1 above.

We start with the following result, which is a consequence of Proposition7.9.

7.3 Minimal actions and bimodule categories 131

Lemma 7.12. Let σ : G y M be a minimal action of the compact groupG on the II1 factor M . Then, the inclusion MG ⊂M is quasi-regular.

Proof. Because of Proposition 7.9, L(π) is a finite index MG-MG-subbi-module of L2(M). It follows that the closure of MGaMG is a finite indexMG-MG-subbimodule of L2(M), for every a ∈ L0(π). Since MG ⊂ M isirreducible, Lemma 6.3 implies that L0(π) ⊂ QNM (MG). We concludeusing Proposition 7.6.

In the following proposition, we construct, for every irreducible represen-tation π ∈ Irr(G), an eigenmatrix Vπ. Such eigenmatrices provide a usefultool to study the spectral subspace of π. For another construction of sucheigenmatrices, see for example [60, Theorem 12 and following comments].

Proposition 7.13. Let σ : G y (M, τ) be a minimal action of the com-pact group G on the II1 factor M . Let π : G → U(Hπ) be an irreduciblerepresentation of G. Then, there exists a unitary Vπ ∈ B(Hπ)⊗M suchthat

(id⊗ σg)(Vπ) = Vπ(π(g)⊗ 1), for all g ∈ G .

Proof. Define F := π ∈ Irr(G) | L0(π) 6= 0.Step 1. We prove that for every π ∈ F , there exists a unitary Vπ ∈B(Hπ)⊗M such that (id⊗ σg)(Vπ) = Vπ(π(g)⊗ 1).

We proceed in a similarly as in the proof of Proposition 7.9. Let π ∈ F .We define

M := M2(C)⊗ B(Hπ)⊗M

with the action β : G y M defined by

βg

(a b

c d

)=(

(id⊗ σg)(a) (id⊗ σg)(b)(π(g)∗ ⊗ 1)(π(g)⊗ 1)(id⊗ σg)(c) (Adπ(g)⊗ σg)(d)

).

The von Neumann algebra M is equipped with the faithful trace τ givenby

τ

(a b

c d

)=

(Tr⊗τ)(a+ d)2 dim(π)

,

where Tr denotes the non-normalized trace on B(Hπ). Let P := Mβ and

p :=(

1 00 0

)and q :=

(0 00 1

).


We prove that P is a factor. By definition of β, we have that(B(Hπ)⊗MG 0

0 1⊗MG

)⊂ P .

By minimality of σ : G y M , we get that

Z(P ) ⊂(

C1 00 B(Hπ)⊗ 1

). (7.7)

Then, every x ∈ Z(P ) yields an element Tx ∈ B(Hπ) such that

x =(

1 00 Tx ⊗ 1

).

By definition of the action β, it follows that Tx ∈ Mor(π, π) = C1, sinceπ ∈ Irr(G). So we have that

Z(P ) ⊂(

C1 00 C1

). (7.8)

Because π ∈ F there exists a non-zero element

t :=(

0 ∗0 0

)∈ P .

Every element in Z(P ) commutes to t, so have Z(P ) = C1, by (7.8).

The projections p and q are equivalent in P , because they have the sametrace. Therefore, there exists a partial isometry V ∈ P such that p = V V ∗

and q = V ∗V . These relations imply that V is of the form(0 Vπ0 0

),

where Vπ is a unitary in B(Hπ)⊗M . Since V ∈ P , the unitary Vπ satisfiesthe required relation.

Step 2. We prove now that F = Irr(G).

In this part of the proof we use the second axiom in the definition ofminimal action saying that the action σ is faithful. Note that so far, wedid not need this fact.

Let π, η ∈ F . We prove that all irreducible sub-representations ρ ⊂ π⊗ ηalso belong to F . Because of Step 1, we have unitaries Vπ and Vη satisfying


respectively (id⊗σg)(Vπ) = Vπ(π(g)⊗1) and (id⊗σg)(Vη) = Vη(η(g)⊗1).Let ρ ∈ Irr(G) such that ρ ⊂ π ⊗ η. Take an isometry v ∈ Mor(ρ, π ⊗ η).Let ξ be a non-zero element of Hπ ⊗Hη and denote

X := (ξ∗ ⊗ 1)(Vπ)13(Vη)23(v ⊗ 1) .

Straightforward computations prove that (id ⊗ σg)(X) = X(ρ(g) ⊗ 1).Furthermore, X is non-zero element of L0(ρ), since (Vπ)13(Vη)23(v⊗ 1) isan isometry. So ρ ∈ F .

Furthermore, since L0(π) = L0(π) (see the proof of Proposition 7.6) wehave that π ∈ F for every π ∈ F .

Recall that we define the coefficient (ξ, η) of the representation π to bethe function satisfying Cπξ,η(g) := 〈π(g)ξ, η〉, for all g ∈ G. Let S ∈Mor(Hπ,M) and ξ ∈ Hπ. Then, for all y ∈M ,

τ(σg(S(ξ))∗y) = τ(S(π(g)ξ)∗y) = Cπξ,S∗(y)(g) .

DefineTF (G) := spanCπξ,η | π ∈ F , ξ, η ∈ Hπ .

It follows from the above results that TF (G) is a ∗-algebra and thus,the norm closure of TF (G) is a C∗-algebra. Since the linear span ofMor(Hπ,M)Hπ is L2-dense in M (see Proposition 7.6), every function

g 7→ τ(σg(x)y), with x, y ∈M

can be approximated in norm by element of TF (G). Let A be the C∗-algebra generated by such functions. Because of the faithfulness of theaction, A separates the points of G. So, the Stone-Weierstrass theoremimplies that A = C(G), the algebra of continuous functions on G. Then,F = Irr(G) hence proving Step 2 and the proposition.

Remark 7.14. From now on, and throughout this chapter, we fix acomplete set Irr(G) of two by two inequivalent and irreducible unitaryrepresentations of G. We also choose, for every π ∈ Irr(G) a unitaryVπ ∈ B(Hπ)⊗M such that

(id⊗ σg)(Vπ) = Vπ(π(g)⊗ 1) .

Note that the unitaries Vπ are uniquely determined up to left multiplica-tion by a unitary element of B(Hπ)⊗MG.


For later use (in the proof of Lemma 10.14), we record the followingelementary property.

Lemma 7.15. Let σ : G y M be a minimal action of the compact groupG on the II1 factor M . Let π and η be irreducible representations of G.Take µ1, . . . , µn ∈ Irr(G), with possible repetitions, and isometries vi ∈Mor(µi, π ⊗ η) satisfying

∑ni=1 viv

∗i = 1. There exist Xi ∈ B(Hµi ,Hπ ⊗

Hη)⊗MG with X∗iXi = 1 for all i and

∑ni=1XiX

∗i = 1 such that

(Vπ)13(Vη)23 =n∑i=1

XiVµi(v∗i ⊗ 1) .

Proof. The decomposition π⊗η = ⊕ni=1µi yields elements vi ∈ Mor(µi, π⊗η) such that v∗i vj = δij1, and

∑ni=1 viv

∗i = 1. We denote by

X := (Vπ)13(Vη)23

(n∑i=1

(vi ⊗ 1)Vµi(v∗i ⊗ 1)

)∗.

We prove that the element X ∈ B(Hπ⊗Hη)⊗M is fixed under the groupG. For all g ∈ G, we have that

(id⊗ id⊗ σg)(X) = (Vπ)13(Vη)23n∑i=1

((π ⊗ η)(g)⊗ 1)(vi ⊗ 1)

(µi(g)∗ ⊗ 1)V ∗µi

(v∗i ⊗ 1)

= (Vπ)13(Vη)23n∑i=1

(vi ⊗ 1)(µi(g)⊗ 1)(µi(g)∗ ⊗ 1)

V ∗µi

(v∗i ⊗ 1)

= X .

We easily check that the Xi := X(vi⊗ 1) are the required isometries.

Lemma 7.16. Let G y M be a minimal action of the compact group Gon the II1 factor M . Let π : G→ U(Hπ) be an irreducible representation.Then, we have

Mor(Hπ,M)Hπ∼= (H∗

π ⊗MG)Vπ(Hπ ⊗ 1) .

Proof. We identify B(Hπ,M) and H∗π ⊗M . The map

H∗π ⊗MG → H∗

π ⊗M : a 7→ aVπ


yields an isomorphism between H∗π ⊗MG and Mor(Hπ,M). Then, the

lemma follows.

We have seen that there is a natural structure of MG-MG-bimodule onthe spectral subspace L(π), for every finite dimensional representation πof G. We prove a well known “Peter-Weyl type” result. Peter-Weyl’s the-ory shows, in particular, that the left regular representation of a compactgroup G is the direct sum of irreducible representations π appearing withmultiplicity dimπ. Letting G act minimally on M , one obtains analogousresults in a von Neumann algebra setting, where finite index bimodulesover the fixed-point algebra MG play the role of unitary representationsof G. It can be proven that the MG-MG-bimodule L2(M) is the directsum of irreducible MG-MG-bimodules indexed by irreducible representa-tions π, each of them appearing with multiplicity dimπ. The unitariesVπ introduced in Proposition 7.13 are the key tools to investigate theseanalogies. Using these unitaries, one builds a map associating a finite in-dex irreducible MG-MG-bimodule to every irreducible representation π.In the next section we prove that this map yields in fact a fully faithfultensor functor Rep(G) → Bimod(MG).

Notation 7.17. From now on, we always denote P := MG, when G actson M minimally.

For every π ∈ Irr(G), we have chosen a unitary Vπ ∈ B(Hπ)⊗M satisfying(id ⊗ σg)(Vπ) = Vπ(π(g) ⊗ 1), for all g ∈ G. We define the unital ∗-homomorphism

ψπ : P → B(Hπ)⊗ P : ψπ(a) = Vπ(1⊗ a)V ∗π (7.9)

Lemma 7.18. Let π be an irreducible representation of G. The Hilbertspace

H(ψπ) := H∗π ⊗ L2(P ) , (7.10)

which is a P -P -bimodule as P(H∗π ⊗ L2(P ))ψπ(P ) is irreducible and, up to

unitary equivalence, independent of the choice of Vπ.

Proof. Take an intertwiner T ∈ MorP (H(ψπ))P = (B(Hπ)⊗P )∩ψπ(P )′.Minimality of the action G y M yields that V ∗

π TVπ ∈ B(Hπ) ⊗ 1. The


element V ∗π TVπ is thus invariant under id⊗σg which implies that V ∗

π TVπ ∈C1, by irreducibility of π. This proves irreducibility of the bimoduleH(ψπ).

Suppose now that there exists an other unitary Ωπ ∈ B(Hπ)⊗P satisfying(id ⊗ σg)(Ωπ) = Ωπ(π(g) ⊗ 1). Denote ψπ(·) := Ωπ(1 ⊗ · )Ω∗

π. We canfind a unitary X ∈ B(Hπ) ⊗ P such that Ωπ = XVπ and thus, ψπ =(AdX) ψπ.

Lemma 7.19. Let G y (M, τ) be a minimal action of the compact groupG on the II1 factor M and π ∈ Irr(G). Then,

Eψπ(P )(x) = ψπ((tr⊗id)(V ∗

π xVπ)), for all x ∈ B(Hπ)⊗ P .

Here, tr denote the normalized trace on B(Hπ). As a consequence, wehave that

〈x, y〉P = (tr⊗id)(V ∗π x

∗yVπ), for all x, y ∈ H(ψπ)0 .

Proof. By uniqueness of the (tr⊗τ)-preserving conditional expectationEψπ(P ) on B(Hπ)⊗P , we only have to prove that for every x ∈ B(Hπ)⊗Pand y ∈ P ,(

tr⊗τ)(ψπ((tr⊗id)(V ∗

π xVπ))ψπ(y)

)= (tr⊗τ)(xψπ(y))

and that (tr⊗id)(V ∗π xVπ) ∈ P , for all x ∈ B(Hπ)⊗ P . We compute(

tr⊗τ)(ψπ((tr⊗id)(V ∗

π xVπ))ψπ(y)

)= τ

((tr⊗id)(V ∗

π xVπ)y), by uniqueness of the trace

= τ((

tr⊗id)(V ∗π xVπ(1⊗ y)

))=(tr⊗τ

)(V ∗π xVπ(1⊗ y)

)= (tr⊗τ)(xψπ(y)) .

We prove now that that (tr⊗id)(V ∗π xVπ) ∈ P , for all x ∈ B(Hπ)⊗ P .

σg((tr⊗id)(V ∗

π xVπ))

= (tr⊗id)((id⊗ σg)(V ∗

π xVπ))

= (tr⊗id)((π(g)∗ ⊗ 1)(V ∗

π xVπ)(π(g)⊗ 1))

= (tr⊗id)((V ∗π xVπ)) .

The last assertion of the lemma follows from Lemma 5.7.


Lemma 7.20. Let σ : G y (M, τ) be a minimal action of the compactgroup G on the II1 factor M . Let π be a finite dimensional representationof G. Take a unitary V ∈ B(Hπ)⊗M such that (id⊗σg)(V ) = V (π(g)⊗1),for all g ∈ G. Define a map

ψ : P → B(Hπ)⊗ P : x 7→ V (1⊗ x)V ∗ .

Then, we have the following isomorphism of P -P -bimodules:

PMor(Hπ,L2(M))P ∼= P

((H∗

π ⊗ L2(P ))ψ(P ) .

Note that the previous lemma implies in particular that H(ψπ) andPMor(Hπ,L2(M))P are isomorphic as P -P -bimodules.

Proof. We define the map

Θ : H∗π ⊗ P → PMor(Hπ,L2(M))P : x 7→ Θ(x) ,

such that Θ(x)ξ := xVπ(ξ ⊗ 1), for all ξ ∈ Hπ.

Denote (ei) the canonical orthonormal basis of Hπ. Then, for all x, y ∈H∗π ⊗ P and for all i,

〈Θ(x),Θ(y)〉 =∑i

τ((e∗i ⊗ 1)V ∗

π x∗yVπ(ei ⊗ 1)

)= τ

((Tr⊗id)

(V ∗π x

∗yVπ))

= (Tr⊗τ)(V ∗π x

∗yVπ)

= (Tr⊗τ)(x∗y) .

This proves that Θ is an isometry on H∗π ⊗ P . Lemma 7.16 implies that

Im(Θ) = Mor(Hπ,M). Then, Θ extends to the required unitary, usingLemma 7.8.

Remark 7.21. Let G y M be a minimal action of the compact groupG on the II1 factor M and denote P := MG. Let π ∈ Irr(G). Then themap AdV ∗

π yields an isomorphism of finite index subfactors(ψπ(P ) ⊂ B(Hπ)⊗ P

) ∼= (1⊗ P ⊂ B(Hπ)⊗ P).

Then, [B(Hπ)⊗ P : ψπ(P )] = (dimπ)2. Then, Lemma 7.20 and Proposi-tion 5.1 imply that

dim(PMor(Hπ,L2(M)

)= dim

(Mor(Hπ,L2(M)P

)= dimπ .


Note that in Proposition 7.9 we have proven that both dimensions aresmaller than dimπ, when the action is not required to be minimal.

Proposition 7.22. Let σ : G y (M, τ) be a minimal action of thecompact group G on the II1 factor M and denote P := MG, the fixed-point algebra. Let π be an irreducible representation of G and consideron B(Hπ)⊗ P the scalar product given by Tr⊗τ . Then, the map

θπ : 1⊗ P

(B(Hπ)⊗ P

)ψπ(P ) → PL

0(π)P : a 7→ dim(π)1/2(Tr⊗id)(aVπ)

satisfies the following properties.

• θπ is P -P -bimodular, bijective and extends to an isometry B(Hπ)⊗L2(P ) → L2(M).

• The adjoint of θπ is given by Eπ := θ∗π satisfying

Eπ(b) = dim(π)1/2∫G(π(g)∗ ⊗ σg(b))V ∗

π dg , (7.11)

for all b ∈M .

• E∗πEπ projects onto L(π) and

∑π∈Irr(G)E

∗πEπ = 1.

Corollary 7.23. Let G y (M, τ) be a minimal action of the compactgroup G on the II1 factor M . Then

L2(M) =⊕

π∈Irr(G)

L(π) .

Moreover, for every irreducible representation π we have L(π) ∼= Hπ ⊗H(ψπ).

Proof of Proposition 7.22. It is easily checked that the map θπ is P -P -bimodular. We prove that θπ is a bijection. Recall (see Lemma 7.16) thatL0(π) is densely spanned by the set

(ξ∗ ⊗ a)Vπ(η ⊗ 1) | ξ, η ∈ Hπ, a ∈ P .

Let ξ, η ∈ Hπ and a ∈ P . Then,

θπ(ηξ∗ ⊗ a) = dim(π)1/2(Tr⊗id

)((ηξ∗ ⊗ a)Vπ

)= dim(π)1/2

(Tr⊗id

)((ξ∗ ⊗ a)Vπ(η ⊗ 1)

),


proving surjectivity of θπ. We prove that θπ is an isometry. Let a, b ∈B(Hπ)⊗ P . We use matrix notations and write

(Tr⊗id)(aVπ) =∑i,j

aij(Vπ)ji .

Then, we have that

τ((Tr⊗id)(aVπ)∗(Tr⊗id)(bVπ)

)=∑i,j,k,l

τ((V ∗π )ija∗ijbkl(Vπ)lk

)=∑i,j,k,l

τ(a∗ijbkl(Vπ)lk(V

∗π )ij

)=∑i,j,k,l

τ(a∗ijbkl EP

((Vπ)lk(V ∗

π )ij))

.

We compute

EP((Vπ)lk(V ∗

π )ij)

=∫G

(Vπ(π(g)⊗ 1)

)lk

(((Vπ(π(g)⊗ 1)

)∗)ijdg

=∑s,t

∫G(Vπ)ls(π(g)⊗ 1)sk(π(g)∗ ⊗ 1)it(V ∗

π )tj dg

=δik

dimπ

∑s,t

(Vπ)ls(V ∗π )tjδst,by Theorem 7.3

=δik

dimπ

∑s

(Vπ)ls(V ∗π )sj

=1

dimπδikδlj .

So, we obtain that

τ((Tr⊗id)(aVπ)∗(Tr⊗id)(bVπ)

)=

1dimπ

∑i,j

τ(a∗ijbij)

=1

dimπ

∑i,j

τ((a∗)ji bij)

=1

dimπ(Tr⊗τ)(a∗b) ,

which proves that θπ is an isometry.

We prove that the adjoint of θπ is given by

Eπ(b) = dim(π)1/2∫G(π(g)∗ ⊗ σg(b))V ∗

π dg .


Note first that by invariant of the Haar measure, the element∫G(π(g)∗⊗

σg(b))V ∗π dg is invariant under id ⊗ σg and thus belongs to B(Hπ) ⊗ P .

Furthermore, we have, for all b ∈M and a ∈ B(Hπ ⊗ P )

dim(π)−1/2 τ(θπ(a)b∗) = τ((Tr⊗id)(aVπ)b∗

)= τ

((Tr⊗id)(aVπ(1⊗ b∗)

)=

∫G

(τ σg

)((Tr⊗id)(aVπ(1⊗ b∗)

)dg

=∫Gτ((

Tr⊗id)(id⊗ σg)(aVπ(1⊗ b∗)

)dg

=∫Gτ((

Tr⊗id)(aVπ(π(g)⊗ σg(b)∗)

))dg

= (Tr⊗τ)(a

(∫G(π(g)∗ ⊗ σg)V ∗

π dg

)∗).

Denote pπ := E∗πEπ. Let x = (ξ∗⊗ a)Vη(eηj ⊗ 1), with ξ ∈ Hη, a ∈ P and

(eηj ) an orthonormal basis of Hη. Elements of this form densely span M ,by Proposition 7.6. We claim that

pπ(x) =x if π = η

0 if π 6= η

This proves that pπ is the orthogonal projection from L2(M) onto L(π)and that the pπ are two by two orthogonal, with

∑π pπ = 1. We prove


our claim.

pπ(x)

= dim(π)1/2(Tr⊗id)(Eπ(x)Vπ)

= dim(π)1/2∑i

((eπi )∗ ⊗ 1)(Eπ(x)Vπ)(eπi ⊗ 1)

= dim(π)∑i

∫Gσg(x)((eπi )

∗ ⊗ 1)(π(g)∗ ⊗ 1)(eπi ⊗ 1) dg

= dim(π)∑i

(ξ∗ ⊗ a)Vη∫G(η(g)⊗ 1)(eηj (e

πi )∗ ⊗ 1)(π(g)∗ ⊗ 1)(eπi ⊗ 1) dg,

because of orthogonality relations; see Lemma 7.2

=∑i

δπ,η(ξ∗ ⊗ a)Vη Tr(eηj (eπi )∗)(eπi ⊗ 1)

=∑i

δπ,ηδi,j(ξ∗ ⊗ a)Vη(eπi ⊗ 1)

= (ξ∗ ⊗ a)Vη(eηj ⊗ 1)δπ,η .

Proof of Corollary 7.23. The orthogonal decomposition of L2(M) imme-diately follows from Proposition 7.22. By definition L0(π) = Mor(Hπ,M)Hπ. So the isomorphism L(π) ∼= Hπ ⊗H(ψπ) follows from Lemma 7.20.In fact, using the description of L0(π) given by Lemma 7.16, it is alsoeasy to prove directly that

Θ : Hπ ⊗H∗π ⊗ L2(P ) → L(π) : η ⊗ ξ 7→ ξVπ(η ⊗ 1)

is the required unitary.

We conclude this section with the following crucial theorem. As ex-plained in section 6.2, this theorem, yielding a fully faithful tensor functorRep(G) → Bimod(P ) goes back to [47] in the study of algebraic quantumfield theory.

Theorem 7.24. Let σ : G y (M, τ) be a minimal action of the compactgroup G on the II1 factor M . Set P := MG. Then,

F : Rep(G) → Bimod(P ) : π 7→ PMor(L2(M),Hπ)P


defines a fully faithful tensor functor from the category Rep(G) of finitedimensional unitary representations of G to the category Bimod(P ) offinite index P -P -bimodules.

In the following proof we assume that our categories are strict. Theassumption is not restrictive since every C∗-tensor category is equivalentto a strict one; see for example [28] where this result is proven for monoidalcategories.

Proof. We fix some notations, before starting the proof. For every finitedimensional representation π, we can take unitaries Vπ ∈ B(Hπ) ⊗ M

such that (id ⊗ σg)(Vπ) = Vπ(π(g) ⊗ 1). We have already constructedsuch unitaries Vπ, for irreducible π. When π is no longer irreducible, wecan decompose it as a direct sum of irreducibles πi. So, we take isometriesvi : Hπi → Hπ such that

∑ni=1 viv

∗i = 1 an then the element

Vπ :=n∑i=1

(vi ⊗ 1)Vπi(v∗i ⊗ 1) (7.12)

is the unitary satisfying the required formulas. Similarly, we also have∗-homomorphisms ψπ : P → B(Hπ)⊗ P such that ψπ(x) = Vπ(1⊗ x)V ∗

π .

We start by proving that F is a tensor functor. We define F on theintertwiners of Rep(G) as follows. For every i ∈ Mor(π, ρ) we define(F (i)(S)

)(ξ) := (iS)(ξ), for every S ∈ PMor(L2(M),Hπ)P , ξ ∈ L2(M) .

(7.13)Then, it is clear that F send the identity map Hπ → Hπ to the identitymap S 7→ S. Moreover, if i ∈ Mor(η, π) and j ∈ Mor(ρ, η), we have thatF (i j) = F (i) F (j). So F is a functor.

Now we prove that F (i∗) = F (i)∗, for every i ∈ Mor(π, η). For everyS ∈ PMor(L2(M),Hπ)P and T ∈ PMor(L2(M),Hη)P we have that

〈F (i)S, T 〉 = τ((F (i)S)∗T )

= τ(S∗i∗T )

= 〈S, F (i∗)(T )〉 .

We prove that we have a natural isomorphism F (π⊗η) ∼= F (π)⊗F (η), forevery π, η ∈ Rep(G). Since PMor(L2(M),Hπ)P and PMor(Hπ,L2(M))P


are isomorphic as P -P -bimodules, it is enough to prove that for everyπ, η ∈ Rep(G) the map

uπ,η : Mor(Hπ,M)Mor(Hη,M) → Mor(Hπ⊗η,L2(M)) ,

defined by uπ,η(T ⊗S)(ξ⊗ ξ′) = T (ξ)S(ξ′) is an isometry that extends toa natural isomorphism. Take (ei) and (fj) orthonormal basis of Hπ andHη. By (7.5), we obtain that

〈uπ,η(T1 ⊗ S1), uπ,η(T2 ⊗ S2)〉

=∑i,j

τ(S1(fj)∗T1(ei)∗T2(ei)S2(fj)

)=∑j

τ(S1(fj)∗〈T1, T2〉PS2(fj)

), by Lemma 7.8

= 〈S1, 〈T1, T2〉P · S2〉 ,

proving injectivity of the map uπ,η. We prove now that

dim1/2

(PMor(Hπ,L2(M))⊗PMor(Hη,L2(M))P

)= dim1/2

(PMor(Hπ⊗η,L2(M))P

).

Decompose π into irreducible representations as π = ⊕iπi. Then, weobtain that

dim1/2

(PMor(Hπ,L2(M))P

)=∑i

dim1/2

(PMor(Hπi ,L

2(M))P)

by Theorem 5.32

=∑i

dim1/2

(P(H∗

πi⊗ L2(P ))ψπi (P )

), by Lemma 7.20

=∑i

(dim

(P(H∗

πi⊗ L2(P ))

)dim

((H∗

πi⊗ L2(P ))ψπi (P )

))1/2,

by Theorem 5.32

=∑i

√(dimπi)2 by Remark 7.21

= dimπ .

This proves the required formula, the map uπ,η is thus an isomorphism.Since it is easy to check that uπ,η is natural in π and η, we have proventhat the functor F is now a tensor functor.


We prove now that F is fully faithful so we prove that the map

Mor(π, η) → MorP (F (π), F (η))P : Ω 7→ F (Ω)

is an isomorphism, for every π, η ∈ Rep(G). We first prove that

Mor(π, η) ∼= MorP (H(ψπ),H(ψη))P , for every π, η ∈ Rep(G) .

Let T ∈ MorP (H(ψπ),H(ψη))P and denote n = dim(π), m = dim(η).Corollary 5.13 and remarks preceding it yield an element

v ∈ Cm(Cn)∗ ⊗ P such that ψη(a)v = vψπ(a), for all a ∈ P . (7.14)

Then, the element V ∗η vVπ commutes with 1⊗a, for all a ∈ P . Minimality

of G y M yields an element

Ω(v) ∈ Cm(Cn)∗ such that v = Vη(Ω(v)⊗ 1)V ∗π . (7.15)

So, in particular, V ∗η vVπ is invariant under the action of G and thus

Ω(v) ∈ Mor(π, η). This procedure shows how to build an intertwinerof π and η, starting from an intertwiner of the bimodules H(ψπ) andH(ψη). The elements v defined in (7.14) are in one to one correspondencewith elements of MorP (H(ψπ),H(ψη))P , by Corollary 5.13, so the abovecomputations can be reversed in the following way, yielding a one toone correspondence between Mor(π, η) and MorP (H(ψπ),H(ψη))P . Wedenote by Rv∗ the operator of right multiplication by v∗ and define themaps

MorP (H(ψπ),H(ψη))P

T 7→v(T ), given by Corollary 5.13


Cm(Cn)∗ ⊗ P

v 7→Ω(v), defined in (7.15)

Cm(Cn)∗ ⊗ P

v 7→Rv∗

OO

Mor(π, η) Mor(π, η)

Ω 7→Vη(Ω⊗1)V ∗π

OO

These maps are obviously each other’s inverse, since Rv(T )∗ = T , byconstruction of v(T ). It is now an exercise to check that the map Ω 7→F (Ω) can be reconstructed using the right map above. See Appendix Afor the precise computations.

7.4 A biduality result 145

7.4 A biduality result

Proposition 7.25. Let σ : G y M be a minimal action of the compactgroup G on the II1 factor M . Denote P := MG. Let α ∈ Aut(M) suchthat α(x) = x, for all x ∈ P . Then, there exists g ∈ G such that α = σg.

Proof. In the proof of Theorem 7.24 we have proven that the unitariesVπ can also be defined for non necessarily irreducible finite dimensionalunitary representations and still satisfy the relation (id ⊗ σg)(Vπ) =Vπ(π(g) ⊗ 1). Let π be a finite dimensional representation of G. Weprove that

(id⊗ α)(Vπ) ∈ Vπ(B(Hπ)⊗ 1

). (7.16)

Let x ∈ P . Then,

(id⊗ α)(Vπ)(1⊗ x) = (id⊗ α)(Vπ(1⊗ x))

= (id⊗ α)(ψπ(x)Vπ)

= ψπ(x)(id⊗ α)(Vπ) .

From these equalities, we conclude that

V ∗π (id⊗ α)(Vπ) ∈ B(Hπ)⊗ (P ′ ∩M) = B(Hπ)⊗ 1 ,

by minimality. Formula (7.16) is proven and yields a unitary Ω(π) ∈B(Hπ) such that

(id⊗ α)(Vπ) = Vπ(Ω(π)⊗ 1) . (7.17)

We have to prove the existence of an element g ∈ G, independent of π,such that Ω(π) = π(g). Then, id⊗ α and id⊗ σg agree on Vπ, by (7.17).So α and σg coincide on L0(π), which is linearly spanned by products ofP and the coefficients of Vπ (see Lemma 7.16). Since spanL0(π) | π ∈Rep(G) generates M , the proposition will be proven.

Note that the element Ω(π) is independent of the choice of Vπ, in thefollowing sense. Whenever there is a W ∈ B(Hπ) ⊗M satisfying (id ⊗σg)(W ) = W (π(g)⊗1), we have (id⊗α)(W ) = W (Ω(π)⊗1). The elementΩ(π) is also uniquely determined by the fact that α S = S Ω(π), forevery S ∈ Mor(Hπ,M). To prove this, we identify S ∈ Mor(Hπ,M) withQS ∈ H∗

π ⊗M . Since (id⊗ σg)(QS) = QS(π(g)⊗ 1), the above reasoningyields the required formula. We prove that the operation Ω also satisfies

Ω(η)v = vΩ(π), for every v ∈ Mor(π, η) , (7.18)


Ω(π) = Ω(π) , (7.19)

Ω(π ⊗ ρ) = Ω(π)⊗ Ω(ρ) . (7.20)

Let v ∈ Mor(π, η). Take S ∈ Mor(Hη,M). Then, α S = S Ω(η). So,α (S v) = S (Ω(η) v). Since S v ∈ Mor(Hπ,M), we have that

S v Ω(π) = S (Ω(η) v) . (7.21)

Observe that H∗η = spanω S | ω ∈ M∗, S ∈ Mor(Hη,M). Then,

(7.21) implies that vΩ(π) = Ω(η)v.

We prove now that Ω(π) = Ω(π). Define J to be the contragredientoperation on Hπ and J the canonical anti-unitary on L2(M). Take anelement S ∈ Mor(Hπ,M). Recall from the proof of Proposition 7.6 thatMor(Hπ,M) = J Mor(Hπ,M)J . So JSJ ∈ Mor(Hπ,M). Then, α S =S Ω(π). Then, we obtain

α JSJ = JSJ J ∗Ω(π)J .

By definition J ∗Ω(π)J = Ω(π). It follows that Ω(π) = Ω(π).

Let π, ρ ∈ Rep(G). We prove now that Ω(π⊗ρ) = Ω(π)⊗Ω(ρ). We havethat

(id⊗ α)((Vπ)13(Vρ)23

)= (id⊗ α)

((Vπ)13

)(id⊗ α)

((Vρ)23

)= (Vπ)13(Ω(π)⊗ 1⊗ 1)(1⊗ VρΩ(ρ)⊗ 1)

=((Vπ)13(Vρ)23

)(Ω(π)⊗ Ω(ρ)⊗ 1) .

The required formula follows from the uniqueness of the element Ω(π⊗ρ).

We recall that the coefficients of π are the functions defined by Cπξ,η(g) :=〈π(g)ξ, η〉; see section 7.1. The coordinate functions Cπξk,ξj form a basisof T (G). Let π ∈ Irr(G). We choose an orthonormal basis (ξi) of Hπ andwe define a linear functional ϕ on the trigonometric polynomials T (G) asfollows.

ϕ : T (G) → C : ϕ(Cπξk,ξj ) = 〈Ω(π)ξk, ξj〉 .

We prove that the linear functional ϕ is multiplicative on T (G). Takeanother irreducible representation ρ and choose an orthonormal basis (ηi)

7.4 A biduality result 147

of Hρ. Decompose π⊗ ρ into irreducibles as π⊗ ρ =⊕r

i=1 µi. So we takeisometries vi ∈ Mor(µi, π⊗ρ) satisfying

∑ri=1 viv

∗i = 1. We first compute

Cπξk,ξj (g)Cρηm,ηl

(g) = 〈π(g)ξk, ξj〉〈ρ(g)ηm, ηl〉

= 〈(π ⊗ ρ)(g)(ξk ⊗ ηm), ξj ⊗ ηl〉

=r∑s=1

〈(π ⊗ ρ)(g)vsv∗s(ξk ⊗ ηm), ξj ⊗ ηl〉

=r∑s=1

〈µs(g)v∗s(ξk ⊗ ηm), v∗s(ξj ⊗ ηl)〉

=r∑s=1

Cµs

v∗s (ξk⊗ηm),v∗s (ξj⊗ηl)(g) .

So, we have that

ϕ(Cπξk,ξjCρηm,ηl

)

=r∑s=1

〈Ω(µs)(v∗s(ξk ⊗ ηm)

), v∗s(ξj ⊗ ηl)〉

=r∑s=1

〈vsΩ(µs)(v∗s(ξk ⊗ ηm)

), ξj ⊗ ηl〉

= 〈Ω(π ⊗ ρ)(ξk ⊗ ηm), ξj ⊗ ηl〉, by (7.18)

= 〈(Ω(π)⊗ Ω(ρ)

)(ξk ⊗ ηm), ξj ⊗ ηl〉, by (7.20)

= 〈Ω(π)ξk, ξj〉〈Ω(ρ)ηm, ηl〉= ϕ(Cπξk,ξj )ϕ(Cρηm,ηl

) .

Multiplicativity of ϕ is proven.

We prove now that ϕ(f) = ϕ(f), for all f ∈ T (G). We first compute

Cπξk,ξj = 〈ξj , π(g)ξk〉

= 〈π(g)ξk, ξj〉= Cπ

ξk,ξj.

So, we obtain

ϕ(Cπξk,ξj ) = 〈Ω(π)ξk, ξj〉

= 〈Ω(π)ξk, ξj〉, by (7.19)

= 〈Ω(π)ξk, ξj〉 .


So ϕ is a multiplicative linear functional on T (G) satisfying ϕ(f) = ϕ(f),for all f ∈ T (G). Then, [19, Theorem 30.5] yields an element g ∈ G suchthat ϕ is the evaluation function at the point g. Then

〈Ω(π)ξk, ξj〉 = 〈π(g)ξk, ξj〉, for all i, j .

This ends the proof.

Part III

Computations of outer

automorphism groups and

bimodules categories of

amalgamated free product

II1 factors

Chapter 8

Amalgamated free product

II1 factors

8.1 Definition and basic properties

We recall basic facts and notations about amalgamated free products andrefer to [39] and [57] for more details.

Definition 8.1. Let M be a von Neumann algebra and M0,M1 ⊂M vonNeumann subalgebras containing a common von Neumann subalgebraN . We suppose that M is endowed with normal conditional expectationE : M → N . The subalgebras M0 and M1 are free with amalgamationover N with respect to E if

E(x1 · · ·xn) = 0 whenever xj ∈Mij such that E(xj) = 0 ,

for every i1 6= i2, . . . , in−1 6= in, where ij ∈ 0, 1, for all j = 1, . . . , n.

Let (M0, τ0) and (M1, τ1) be von Neumann algebras endowed with thefaithful normal traces τ0 and τ1. Let N be a common von Neumannsubalgebra such that τ0|N = τ1|N . We construct the following.

• A Hilbert space H and normal representations λi of Mi on H,

• A faithful normal trace τ on the von Neumann algebra

M := λ0(M0) ∪ λ1(M1)′′ ⊂ B(H) ,

152 Chapter 8. Amalgamated free product II1 factors

such that λ0(M0) and λ1(M1) are free with amalgamation over N , withrespect to the unique τ -preserving conditional expectation E : M → N .The von Neumann algebra M is called the amalgamated free product ofM0 and M1 over N and denoted M := M0 ∗N M1.

The GNS construction for M0 and M1 yields the following data, for i ∈0, 1.

• A Hilbert space Hi := L2(Mi) .

• A normal representation πi : Mi → B(Hi) .

• A unit cyclic and separating vector ξi such that τi(a) = 〈ξi, πi(a)ξi〉,for all a ∈Mi .

We defineHi := Hi Nξi. Fix a unit vector ξ and set

H := Nξ ⊕⊕n≥1

⊕i1 6=i2,...,in−1 6=in

Hi1 ⊗N · · · ⊗N

Hin

. (8.1)

We want to represent M0 and M1 on H. We re-write H in two differentways. The following equalities are only written to get a better under-standing of the construction of the unitary Vi defined further.

H = (Nξ ⊕H1)⊕

H2 ⊕ (

H1 ⊗N

H2)⊕ . . .

= H1 ⊕H2 ⊕ (

H1 ⊗N

H2)⊕ . . .

∼= H1 ⊕ (Nξ ⊗NH2)⊕ (

H1 ⊗N

H2)⊕ . . .

∼= H1 ⊕((Nξ ⊕

H1)⊗N

H2

)⊕ . . .

= H1 ⊕ (H1 ⊗NH2)⊕ . . .

...∼= H1 ⊗N

(Nξ ⊕

H2 ⊕ (

H2 ⊗N

H1)⊕ . . .

).

And similarly,

H ∼= H2 ⊗N(Nξ ⊕

H1 ⊕ (

H1 ⊗

H2)⊕ . . .

).

8.1 Definition and basic properties 153

More precisely, defining, for i = 0, 1,

H(l, i) := Nξ ⊕⊕n≥1

⊕i1 6=i; i1 6=i2,...,in−1 6=in, in

Hi1 ⊗N · · · ⊗N

Hin

,

the map Vi : Hi ⊗N H(l, i) → H such that

Nξi ⊗N Nξ 7→ Nξ ,Hi ⊗N Nξ 7→

Hi ,

Nξi ⊗N( Hi1 ⊗N · · · ⊗N

Hin

)7→

Hi1 ⊗N · · · ⊗N

Hin ,

Hi ⊗N

( Hi1 ⊗N · · · ⊗N

Hin

)7→

Hi ⊗N

Hi1 ⊗N · · · ⊗N

Hin ,

is a unitary. So if we let

λi : B(Hi) → B(H) : a 7→ Vi(a⊗ 1H(l,i))V∗i ,

we obtain representations λi πi of Mi on H. Since the traces τi arefaithful, the GNS representations πi are also faithful. So, we may viewλi as a representation of Mi on H.

We have a right analogue of the previous construction. We define, fori = 0, 1,

H(r, i) := Nξ ⊕⊕n≥1

⊕i1 6=i2,...,in−1 6=in, in, in 6=i

Hi1 ⊗N · · · ⊗N

Hin

.

We define, similarly, the unitary operator Wi : H(r, i) ⊗N Hi → H suchthat

Nξ ⊗N Nξi 7→ Nξ ,

Nξ ⊗NHi 7→

Hi ,(

Hi1 ⊗N · · · ⊗NHin

)⊗N Nξi 7→

Hi1 ⊗N · · · ⊗N

Hin ,(

Hi1 ⊗N · · · ⊗NHin

)⊗N

Hi 7→

Hi1 ⊗N · · · ⊗N

Hin ⊗N

Hi .

Then,ρi : B(Hi) → B(H) : a 7→Wi(1H(r,i) ⊗ a)W ∗

i

defines a representation of B(Hi) onto H. Composing ρi with the canon-ical anti-representation on Hi gives an anti-representation of Mi on H.As for λi, we may view ρi as an anti-representation of Mi on H.


Definition 8.2. Let (M0, τ0) and (M1, τ1) be tracial von Neumann alge-bras with a common von Neumann subalgebra N such that τ0|N = τ1|N .The amalgamated free product of (M0, τ0) and (M1, τ1) over N is the vonNeumann algebra acting on the Hilbert space H (see (8.1)) defined by

M0 ∗N M1 := λ0(M0) ∪ λ1(M1)′′ .

Proposition 8.3. Let (M0, τ0) and (M1, τ1) be von Neumann algebraswith a common von Neumann subalgebra N such that τ0|N = τ1|N . LetM := M0 ∗N M1. Define the vector state τ(x) := 〈ξ, xξ〉, for all x ∈ M .Then, we have the following.

1. For every i1 6= i2 . . . 6= in, with ij ∈ 0, 1, we have

τ(λi1(a1) · · ·λin(an)

)= 0, for all aj ∈ ker(τij ) . (8.2)

2. The vector ξ is cyclic .

Proof. Note that if τ(aj) = 0 then 〈ξij , ajξij 〉 = τj(aj) = 0 so the element

ajξij belongs toHij . So we have that

λij (aj)ξ = Vj(aj ⊗ 1)V ∗j ξ

= Vj(ajξij ⊗ ξ)

= ajξij .

Let aj ∈ ker(τij ), for all j = 1, . . . , n. By induction, we obtain that

λi1(a1) · · ·λin(an)ξ = a1ξi1 ⊗N · · · ⊗N anξin ∈Hi1 ⊗N · · · ⊗N

Hin , (8.3)

which is orthogonal to ξ. Point 1 is proven.

Since ξij is a cyclic vector in Hij , we have that (Mj N)ξij is dense inHij . So, by (8.3) ξ follows cyclic.

Corollary 8.4. Let N,M0,M1 as in the foregoing proposition. The GNSstate τ := 〈ξ, · ξ〉 satisfies the following.

• τ λi = τi, for i = 0, 1 .

• τ is a faithful trace .


Moreover, there exists a conditional expectation E : M → N satisfying

τ = τ0 E = τ1 E .

The following holds.

• E λi = Ei, where Ei denotes the τi-preserving conditional expecta-tion Ei : Mi → N .

• The von Neumann algebras λ0(M0) and λ1(M1) are free with amal-gamation over N with respect to E .

Proof. Let x ∈Mi, i = 0, 1. Proposition 8.3 now implies that

τ(λi(x)

)= τ

(λi((x− τi(x)1)

)+ τi(x)1

)= τi(x)‖ξ‖2 = τi(x) .

We prove that τ is a trace. Remark that the representation λi and theanti-representation ρi constructed above commute. So, H is an Mi-Mi-bimodule, with left and right actions given by λi and ρi. An immediatecomputation yields that a · ξ = ξ · a, for all a ∈ Mi. So we have, for alla ∈Mi and x ∈M := λ0(M0), λ1(M1)′′,

τ(ax) = 〈ξ, axξ〉= 〈a∗ξ, xξ〉= 〈ξa∗, xξ〉= 〈ξ, xξa〉= 〈ξ, xaξ〉 .

So τ(ax) = τ(xa) for all a ∈ Mi and x ∈ M . Then, by induction on thelength of reduced words in M , this holds true for all a ∈M as well.

We prove now that τ is faithful. We have already noticed that λi andρi commute. In fact, because of Theorem [57, Lemma 1.8], we have thatλ0(M0), λ1(M1)′ = ρ0(M0), ρ1(M1)′′. Similar computations as in theproof of Proposition 8.3 imply that the vector ξ is cyclic for the vonNeumann algebra ρ0(M0), ρ1(M1)′′ as well. Then, ξ is separating. Sothe vector state 〈ξ, ·ξ〉 is faithful.


Let E be the unique τ -preserving conditional expectation E : M → N .Let aj ∈ ker(Eij ) for every i1 6= i2 . . . 6= in, with ij ∈ 0, 1. Since theelement E(a1 · · · an) is uniquely determined by the identity

τ(bE(a1 · · · an)

)= τ(ba1 · · · an),

for every b ∈ N , the freeness with amalgamation of λ0(M0) and λ1(M1)will follow from the fact that τ(ba1 · · · an) = 0. Note that aj ∈ ker(τij )since Eij is preserved by τij . Write ba1 = ba1 − τ1(ba1) + τ(ba1). Then,(8.2) allows us to conclude.

Proposition 8.5. Let (M0, τ0) and (M1, τ1) be von Neumann algebraswith a common von Neumann subalgebra N such that τ0|N = τ1|N . LetEi : Mi → N the unique τi-preserving conditional expectations, for i =0, 1. The amalgamated free product M0 ∗NM1 is, up to E-preserving iso-morphism sending Mi to Mi identically, the unique pair (M,E) satisfyingthe following two conditions.

• The von Neumann algebra M is generated by embeddings Mi ⊂M ,i = 0, 1, and is equipped with a conditional expectation E : M → N

satisfying E|Mi= Ei, for i = 0, 1.

• The subalgebras M0 and M1 are free with amalgamation over Nwith respect to E.

Proof. Consider such a pair (M,E). Note that M is densely spannedby N and reduced words of the form a1 · · · an, where Eij (aj) = 0, fori1 6= . . . 6= in and ij ∈ 0, 1. Denote τM := τ0 E = τ1 E. We provethat τM is a trace on M satisfying τM |M0

= τ0 and τM |M1= τ1. It is

enough to prove that τM defines a trace on the set of reduced words. Letaj , bj ∈ ker(Eij ) for every i1 6= i2 . . . 6= in, with ij ∈ 0, 1. Define thefunctions

φk(b) := Eik(a∗kbbk

), ψk(b) := Eik

(bkba

∗k

),

for all k = 1, . . . , n. Note that

τM(φk(x)y

)= τM

(xψk(y)

), for all x, y ∈ N . (8.4)


Let a := a1 · · · an and b := b1 · · · bm. If n 6= m then, τM (a∗b) = 0. Wesuppose now that n = m and, we have

τM (a∗b)

= τM(a∗n · · · (a∗1b1) · · · bn

)= τM

(a∗n · · ·

((a∗1b1)− Ei1(a

∗1b1)

)· · · bn

)+ τM

(a∗n · · ·Ei1

(a∗1b1

)· · · bn

)= τM

(a∗n · · · a∗2φ1(1)b2 · · · bn

)...

= τM(φn · · · φ1(1)

)= τM

(φn−1 · · · φ1(1)ψn(1)

), by (8.4)

...

= τM(ψ1 · · · ψn(1)

)...

= τM(b1 · · · (bna∗n) · · · a∗1

).

A similar computation as the one above implies that the

U : M0 ∗N M1 →M : λi1(a1) · · ·λin(an) 7→ a1 · · · an ,

where aj ∈ ker(Eij ) for every i1 6= i2 . . . 6= in, with ij ∈ 0, 1 is anisometry and extends to a unitary U : L2(M0 ∗N M1, τ) → L2(M, τM ).The unitary U implements the spatial isomorphism M0 ∗N M1

∼= M .

Let (M0, τ0) and (M1, τ1) be von Neumann algebras with a common vonNeumann subalgebra N such that τ0|N = τ1|N . Suppose that the vonNeumann algebra M generated by M0 and M1 is expected on N . Then,the foregoing proposition gives an easy criterion to prove that M is theamalgamated free product of M0 and M1 over N , we only need to checkthat M0 and M1 are free with amalgamation over N inside M . In thecase of amalgamation over the field C, we call M = M0 ∗M1 the freeproduct of M0 and M1.

Example 8.6. Using Proposition 8.5, we easily obtain the following ex-amples.

• Let Γ0,Γ1 be countable groups containing the subgroup Γ. Then,

L(Γ0 ∗Γ Γ1) ∼= L(Γ0) ∗L(Γ) L(Γ1) .


In particular, we have that L(F2) ∼= L(Z) ∗ L(Z).

• Let Γ0,Γ1 be a countable groups acting outerly on the II1 factor R.Then,

(Ro Γ0) ∗R (Ro Γ1) ∼= Ro (Γ0 ∗ Γ1) .

8.2 Some rigidity results: work of Ioana, Peter-

son, Popa

Many striking results were recently obtained using the deformation/rigi-dity principle. For example, in [21] A.Ioana, J.Peterson and S.Popa applythis principle to study amalgamated free products II1 factors. They ob-tained, among many other important results, examples of II1 factors withprescribed abelian compact outer automorphism group. In particular,they proved the existence of II1 factors with trivial outer automorphismgroup, which was a long standing open problem posed by Connes. Thetwo following theorems are the key ingredients for the whole of this thesis.We refer to [21] for the proof, see also Theorems 4.6 and 5.6 in [20] foralternative proofs.

Theorem 8.7 (Theorem. 1.1 in [21]). Let (M0, τ0) and (M1, τ1) be vonNeumann algebras with faithful normal tracial states and a common vonNeumann subalgebra N . Set M = M0 ∗N M1.

Let Q ⊂ M0 be a, possibly non-unital, inclusion with the property thatQ 6≺M0 N . Then every Q-M0-subbimodule H of 1Q L2(M) such thatdim(HM0) <∞ is contained in L2(M0).

In particular, the quasi-normalizer of Q inside 1QM1Q is contained in1QM01Q.

Theorem 8.8 (Theorem. 4.3 in [21]). Let (M0, τ0) and (M1, τ1) be vonNeumann algebras with faithful normal tracial states and a common vonNeumann subalgebra N . Set M = M0 ∗N M1.

If Q ⊂ M is a, possibly non-unital, inclusion and Q ⊂ 1QM1Q has therelative property (T), there exists i ∈ 0, 1 such that Q ≺M Mi.

Combining Theorem 8.7 and Theorem 8.8 we obtain the following corol-lary. The proof of this corollary is inspired from [54, Corollary 2.2]. In

8.2 Some rigidity results: work of Ioana, Peterson, Popa 159

[54], the inclusion Q ⊂ M is not supposed to be quasi-regular, it is re-quired that the quasi-normalizer of Q inside M has finite index in M .

Corollary 8.9. Let (M0, τ0) and (M1, τ1) be von Neumann algebras withfaithful normal tracial states and a common von Neumann subalgebra N .Set M = M0 ∗N M1 and suppose that M0 and M1 are different from N .Suppose that Q ⊂ M is a subfactor such that the inclusion Q ⊂ M isrigid and quasi-regular. Then,

Q ≺M N .

Proof. Suppose that Q 6≺M N . By Theorem 8.8, there exists i ∈ 0, 1such that Q ≺M Mi. Then, there exist a projection p ∈ Mn

i , a non-zero partial isometry v ∈

((Cn)∗ ⊗M

)p and a unital ∗-homomorphism

ψ : Q→ pMni p satisfying xv = vψ(x), for all x ∈ Q. We may also assume

that p is the support of EMi(v∗v), see Remark 5.20. By construction (see

the proof of Theorem 5.19) the Q-Mi-bimodule

ψ(Q)

(p(Cn ⊗ L2(Mi)

))Mi

is isomorphic with a Q-Mi-subbimodule of L2(M). If ψ(Q) ≺Mi N then,there exists a unital ∗-homomorphism η : Q → qNmq and a non-zeropartial isometry w ∈ p

(Cn(Cm)∗ ⊗Mi

)such that ψ(x)w = wη(x), for all

x ∈ Q. Then, xvw = vwη(x), for all x ∈ Q. If we prove that vw is non-zero, we obtain that Q ≺Mi N , contradicting our assumption. Supposethat vw = 0. Then, EMi(v

∗v)w = 0 and thus w = pw = 0, contradiction.So we have proven that ψ(Q) 6≺Mi N . Theorem 8.7 implies that thequasi-normalizer of ψ(Q) inside pMnp is contained in pMn

i p and alsov∗v ∈ pMn

i p. Then we have that

v∗ QNM (Q)v ⊂ QNpMnp(v∗Qv) ⊂ v∗vQNpMnp(ψ(Q)) ⊂ pMn

i p .

By quasi-regularity of the inclusion Q ⊂ M we obtain v∗Mv ⊂ Mni and

thus M = Mi. This is a contradiction since we assumed that M is anon-trivial amalgamated free product.

The following lemma is a crucial ingredient of the proof of Theorem 8.15and also plays an essential role in the proof of our Theorem 9.1. Thislemma is a consequence of Lemma 8.4 in [21] (see also Propositions 3.3and 3.5 in [54]). For convenience, we give a full proof here.


Lemma 8.10. Let Γ0,Γ1 be ICC countable groups. Let Γ0 and Γ1 actouterly on the II1 factors A0 and A1 respectively. Set M := A1 o Γ1

and suppose that α : A0 o Γ0 → A1 o Γ1 is an isomorphism such thatα(A0) ≺M A1 and A1 ≺M α(A0). Then, there exists a unitary u ∈ U(M)such that uα(A0)u∗ = A1.

The following lemma is a key ingredient for the proof of Lemma 8.10, theproof is inspired from the one of Proposition 3.5 in [54] although here, inopposition with Proposition 3.5 in [54], we consider regular inclusions.

Lemma 8.11. Let M be a type II1 factor and A,B ⊂ M regular andirreducible subfactors. We suppose that the following hold.

A ≺M B and B ≺M A .

Then, there exists an A-B-subbimodule K ⊂ AL2(M)B satisfying dim(AK)<∞ and dim(KB) <∞.

Proof. Denote by Tr the trace on the basic construction 〈M, eB〉 andby TrB its restriction to A′ ∩ 〈M, eB〉. Theorem 5.19 yields a non-zeroprojection p1 ∈ A′ ∩ 〈M, eB〉 such that TrB(p1) <∞.

We define

p0 :=∨

p ∈ A′ ∩ 〈M, eB〉 | p 6= 0, TrB(p) <∞. (8.5)

We prove that p0 = 1. Since A ⊂ M is regular, we have that uAu∗ = A,for every unitary u ∈ NormM (A)) so in particular up1u

∗ ∈ A′ ∩ 〈M, eB〉.Then,

p :=∨

u∈NormM (A)

up1u∗ ≤ p0 .

But the projection p is non-zero and, by regularity of A ⊂ M and irre-ducibility of B ⊂M , we have that

p ∈ NormM (A)′ ∩ 〈M, eB〉 = M ′ ∩ 〈M, eB〉= J(M ∩B′)J = C1 .

Then, p0 = p = 1, as claimed. Note that

B′ ∩ 〈M, eA〉 = B′ ∩ JA′J= J(JB′J ∩A′)J= J(〈M, eB〉 ∩A′)J .


Denote by TrA the restriction of the trace Tr on 〈M, eA〉 to B′ ∩ 〈M, eA〉.The above computations show that TrA defines a trace on A′ ∩ 〈M, eB〉.Note that since A′ ∩ 〈M, eB〉 is not a factor, there is a priori no reasonwhy TrA and TrB should coincide but we still have that∨

p ∈ A′ ∩ 〈M, eB〉 | p 6= 0, TrA(p) <∞

= 1 .

Then, we can take non-zero projections p, q ∈ A′ ∩ 〈M, eB〉 such thatTrB(p) < ∞ and TrA(q) < ∞ and pq 6= 0. Define x = pqp ∈ A′ ∩〈M, eB〉+. Then, TrB(x) <∞ and

TrA(x) = TrA(pqp)

= TrA(qpq)

≤ TrA(q) <∞ .

Let δ > 0. Then, the spectral projection r := χ]δ,+∞[(x) defines anelement of A′ ∩ 〈M, eB〉 such that TrB(r) ≤ δ−1 TrB(x) < ∞. Similarly,TrA(r) < ∞. Then, K := r L2(M) is an A-B-subbimodule of AL2(M)Bwith finite left and right dimension.

Now we can prove Lemma 8.10.

Proof of Lemma 8.10. We first prove that it is only necessary to find aunitary u ∈ U(M) such that uα(A0)u∗ ⊂ A1. Indeed, if such unitaryexists, we repeat the proof replacing α by α−1 and thus, we obtain aunitary v ∈M such that vA1v

∗ ⊂ α(A0). If follows that

(uv)A1(uv)∗ ⊂ A1 . (8.6)

We prove that uv ∈ A1. Then, all the inclusions

(uv)A1(uv)∗ ⊂ uα(A0)u∗ ⊂ A1

are equalities. Denote by w := uv. Because of (8.6), we have an auto-morphism β ∈ Aut(A1) such that wx = β(x)w, for all x ∈ A1. Write w asthe L2-convergent sum w =

∑g∈Γ1

wgug. Repeating the proof of Lemma2.17, we obtain that w = we ∈ A1.

The previous lemma yields a finite index inclusion ψ : A0 → pAn1p and apartial isometry

v ∈((Cn)∗ ⊗M

)p, such that α(a)v = vψ(a), for all a ∈ A0 . (8.7)


Since A0 is a factor and ψ(A0) ⊂ pAn1p has finite index, the relativecommutant ψ(A0)′∩pAn1p is finite dimensional; see Lemma 4.12. Cuttingdown ψ by a minimal projection in ψ(A0)′ ∩ pAn1p, we may assume fromthe beginning that ψ is an irreducible inclusion.

The proof now reduces to showing that ψ(A0)′ ∩ pMnp = Cp. Thisimplies that v∗v = p and we show how to conclude the proof. Since Γ0

acts outerly on A0, the inclusion A0 ⊂ A0 o Γ0 is irreducible. Hence,Formula (8.7) implies that vv∗ = 1. By a classical procedure, factorialityof A1 allows us to assume that v ∈ U(M) and that v∗α(A0)v ⊂ A1. Weexplain this fact. Denote by q := e11 ⊗ 1 ∈ Mn(C) ⊗ A1. Then, the twoprojections q and v∗v have same trace. Factoriality of A1 yields a partialisometry w ∈ An1 such that w∗w = v∗v and ww∗ = q. Write

vw∗ =n∑j=1

e∗j ⊗ uj ∈ (Cn)∗ ⊗M .

From equation (8.7) we obtain, for all a ∈ A0,

(vw∗)∗α(a)(vw∗) = wv∗α(a)vw∗

= wv∗vψ(a)w∗

= wψ(a)w∗

= qwψ(a)w∗q

⊂ qAn1q .

The previous inclusion implies that u1 is the only non-zero componentof vw∗ and that u∗1α(A0)u1 ⊂ A1. Moreover, vw∗(vw∗)∗ = vv∗ = 1 sou1u

∗1 = 1. Since we also have (vw∗)∗vw∗ = ww∗ = q we obtain that

u∗1u1 = 1 and u1 is the required unitary.

Recall that M := A1 o Γ1 and denote σ : Γ1 y A1. We still denote by σthe action id ⊗ σ : Γ1 y An1 . Then, we have Mn = An1 o Γ1. Denote byQ := ψ(A0). We prove that Q has trivial relative commutant in pMnp.Let x ∈ Q′∩pMnp. Writing x as the L2-convergent sum x =

∑g∈Γ1

xgug,with xg ∈ pAn1σg(p), one obtains that

yxg = xgσg(y), for all y ∈ Q . (8.8)

Since Q ⊂ pAn1p was assumed to be irreducible, formula (8.8) implies thatxg is a multiple of a unitary in pAn1σg(p). We define the subgroup of Γ1

K := g ∈ Γ1 | There exist xg ∈ pAn1σg(p) such that xgx∗g = p,

x∗gxg = σg(p) and, σg(y) = x∗gyxg, for all y ∈ Q ,


and obtain that Q′ ∩ pMnp ⊂ (A1 o K)n. So, by irreducibility of Q ⊂pAn1p, Q has trivial relative commutant once we have proven that thegroup K is trivial.

We prove that K = e in two steps.

Step 1. K is finite.

We define the embedding

i : K → α ∈ Aut(pAn1p) | α(y) = y, for all y ∈ Q : g 7→ (Adxg) σg .

We prove that this map is well defined. This implies finiteness of the groupK since there are only finitely many automorphisms of a II1 factor Mfixing pointwise a finite index irreducible subfactor N ⊂ M ; see Lemma8.12. We prove that the unitaries xg are unique up to a scalar of modulusone, so in particular, i does not depend on the choice of xg. Suppose thatthere exist another family of unitaries yg, satisfying the same propertiesas xg. Then, the element xgy∗g ∈ pAn1p commutes with Q, so there existsλ ∈ C such that xgy∗g = λp. Since xgx

∗g and ygy

∗g are projections, it

follows that |λ| = 1.

Step 2. We prove that K is a normal subgroup of Γ1 by proving thatΓ1 = ∆, where ∆ is the subgroup of Γ1 given by

∆ := g ∈ Γ1 | There exist xg ∈ pAn1σg(p) such that xgx∗g = p,

x∗gxg = σg(p) and, σg(Q) = x∗gQxg .

The group K is obviously a normal subgroup of ∆. We prove now thatΓ1 = ∆. Let u ∈ NormA0oΓ0(A0). Then, there exists an automorphismβ ∈ Aut(A0) such that

u∗xu = β(x), for all x ∈ A0 . (8.9)

Formula (8.7) and (8.9) imply that

v∗α(u)vψ(β(x)

)= ψ(x)v∗α(u)v, for all x ∈ A0 . (8.10)

Indeed,

v∗α(u)vψ(β(x)) = v∗α(uβ(x)

)v, by (8.7)

= v∗α(xu)v, by (8.9)

= v∗α(x)α(u)v

= ψ(x)v∗α(u)v .


Note that v∗α(u)v ∈ pAn1p. Writing v∗α(u)v as the L2-convergent sumv∗α(u)v =

∑g∈Γ1

xgug, with xg ∈ pAn1σg(p), one obtains that

ψ(x)xg = xg(σg ψ β)(x), for all x ∈ A0, g ∈ Γ1 . (8.11)

Because β is an automorphism of A0 and σg an automorphism of A1,formula (8.11) implies that v∗α(u)v ∈ p(A1 o ∆)np. Then, since A0 ⊂A0 o Γ0 is regular we obtain that

v∗(A1 o Γ1)v ⊂ p(A1 o ∆)np .

Then, A1 o Γ1 = A1 o ∆ and hence, Γ1 = ∆.

Lemma 8.12. Let N ⊂ M be a finite index and irreducible inclusion ofII1 factors. Then,

K := α ∈ Aut(M) | α(x) = x, for all x ∈ N

is a finite group.

Proof. Let α ∈ Aut(M) such that α(x) = x, for all x ∈ N . Note thatif α 6= id, then α is outer. Indeed, if there exists u ∈ U(M) such thatα = Adu, we have that u ∈M ∩N ′ = C1.

For every α ∈ K we define uα : L2(M) → L2(M) such that uα(x) = α(x),for all x ∈ M . Then, uα ∈ U(〈M, eN 〉) and uαxu

∗α = α(x). Since all

α ∈ K \ id are outer, it follows that EM (uα) = 0, for all α ∈ K \ id.In particular, if we denote by Tr the trace on the II1 factor 〈M, eN 〉, wehave that

Tr(uα) = 0, for all α ∈ K \ id . (8.12)

Since [〈M, eN 〉 : M ] = [M : N ] <∞, the inclusion N ⊂ 〈M, eN 〉 also hasfinite index and thus, N ′∩〈M, eN 〉 if finite dimensional; see Lemma 4.12.The map

K → U(〈M, eN 〉 ∩N ′) : α 7→ uα

is injective because of (8.12) and hence, K is finite.

The following lemma is a combination of Theorem 8.7 and Theorem 8.8.


Lemma 8.13. Let M0,M1 be II1 factors with a common subfactor N .We denote M := M0 ∗N M1. Let B ⊂ pMnp be a subfactor with property(T). We suppose that there exits no non-trivial ∗-homomorphism froma II1 factor having property (T) to M1. Then, there exists a projectionq ∈ Mm

0 and an element u ∈ p(Cn(Cm)∗ ⊗ M)q such that uu∗ = p,u∗u = q and

u∗ QNpMnp(B)′′u ⊂ qMm0 q .

Proof. We start by proving the existence of an element u ∈ p(Cn(Cm)∗⊗M)q such that uu∗ = p, u∗u = q and u∗Bu ⊂ qMm

0 q. Let p0 be any non-zero projection in B′ ∩ pMnp. The von Neumann algebra Bp0 still hasproperty (T). Theorem 8.8 implies that Bp0 ≺M Mi for some i ∈ 0, 1.If i 6= 0, Theorem 5.19 yields a non-trivial ∗-homomorphism from Bp0 toM1. Because of our assumptions it follows that i = 0. Theorem 5.19 yieldsa projection q0 ∈ Mm

0 , a non-zero partial isometry v0 ∈ p0(Cn(Cm)∗ ⊗M)q0 and a unital ∗-homomorphism ρ0 : B → q0M

m0 q0 satisfying xv0 =

v0ρ0(x) for all x ∈ B. Since ρ0(B) has property (T), we know thatρ0(B) 6≺M0 N . By Theorem 8.7, it follows that ρ0(B)′ ∩ q0M

mq0 ⊂q0M

m0 q0. In particular, v∗0v0 ∈ q0M

m0 q0. So, we may assume that q0 =

v∗0v0, by Remark 5.19. If v0v∗0 = p0 then, we are done. Suppose thatp1 := 1− v0v∗0 6= 0. Then, since p1 ∈ B′ ∩ pMnp there exists a projectionq1 ∈ Mm

0 , a ∗-homomorphism ρ1 : B → q1Mm0 q1, a non-zero partial

isometry v1 ∈ p1(Cn(Cm)∗⊗M)q1 satisfying v∗1v1 = q1 and xv1 = v1ρ1(x)for all x ∈ B . Then, v1v∗1 is orthogonal to v0v∗0. Now, we consider theset E , ordered by inclusion, and consisting of elements (vi, ρi, qi)i∈I suchthat

• vi ∈ p(Cn(Cm)∗ ⊗M

)qi is a non-zero partial isometry,

• ρi : B → qiMm0 qi is a unital ∗-homomorphism such that xvi =

viρi(x), for all x ∈ B,

• v∗i vi = qi ∈Mm0 ,

• viv∗i are two by two orthogonal.

Zorn’s Lemma yields a maximal family (vi, ρi, qi)i∈I ∈ E . Then,∑i∈I

viv∗i = p .


If it was not the case, r := p −∑

i viv∗i is a non-zero projection in B′ ∩

pMnp. Using once again the fact that Br ≺M M0 we get a projectionq ∈Mm

0 , a ∗-homomorphism ρ : B → qMm0 q, a non-zero partial isometry

v ∈ r(Cn(Cm)∗ ⊗M)q satisfying v∗v = q and xv = vρ(x) for all x ∈ B .

Then, vv∗ ≤ r = p −∑

i viv∗i and thus, vv∗ is orthogonal to every viv

∗i ,

which would contradict the maximality of the family. We can also assumethat the qi are two by two orthogonal, by factoriality of M0. We defineu :=

∑i vi and q :=

∑i qi. Then uu∗ = p, u∗u = q and we have, for all

x ∈ B:

u∗xu =∑i,j

v∗i xvj

=∑i,j

v∗i vjρj(x)

=∑i

qiρi(x)

⊂∑i

qiMm0 qi ⊂ qMm

0 q

Hence,u∗Bu ⊂ qMm

0 q. (8.13)

Since uBu∗ still has property (T), we have that u∗Bu 6≺M0 N . Then,QNqMmq(u∗Bu)′′ ⊂ qMm

0 q, by Theorem 8.7.

Remark 8.14. Note that the foregoing lemma is, as stated, not optimaland was written as such only to fit to our purpose, without seeking formore generality. Indeed, the assumptions on B and M1 can be weakened.It is only necessary to assume rigidity of the inclusion B ⊂ pMnp (whichof course holds if B has property (T) itself) and that B 6≺M M1. See [21,Theorem 5.1].

In the following theorem, Ioana, Peterson and Popa proved that everyabelian compact group can be realized as the outer automorphism groupof a II1 factor, as a consequence they answered a long standing open prob-lem, proving the existence of II1 factors for which every automorphismis inner. Their theorem relies on Theorem 8.7 and Theorem 8.8. Theirresults and the strategy they used to prescribe abelian compact outerautomorphism groups of II1 factors were the starting point for the wholeof the work done in this thesis. Before proving their theorem, we explain


the general ideas of the methods used since these are the methods thatwe generalized to obtain also the non-abelian compact groups.

The setting is the following. They consider an outer action σ of an ICCcountable group Γ on the hyperfinite II1 factor R. The group Γ is a freeproduct of two countable groups Γ0,Γ1 and thus the II1 factor R o Γ isthe amalgamated free product over R of the II1 factors RoΓ0 and RoΓ1.We denote

M := Ro Γ = M0 ∗RM1, with Mi = Ro Γi, i = 0, 1 .

The aim is to prove that under suitable assumptions on the groups Γ0,Γ1

and on their actions on R one obtains that Out(M) ∼= Char(Γ0 ∗Γ1). Letα ∈ Aut(M). We want to prove the existence a character ω ∈ Char(Γ)such that α = αω, up to unitary conjugacy in M ; see notation 2.22.

Step 1. α(R) = R, up to conjugating by a unitary in M .

Then, Proposition 2.19 provides an automorphism δ ∈ Aut(Γ) and uni-taries (xδ(g))g∈Γ ∈ U(R) such that

α(ug) = xδ(g)uδ(g), for all g ∈ Γ . (8.14)

Step 2. Under certain assumptions on the groups Γ0 and Γ1, Kurosh’stheorem applied to the automorphism δ allows us to assume that δ is ofthe form δ = δ0 ∗ δ1, where δi ∈ Aut(Γi), i = 0, 1.

Step 3. Denote β := α|R. Then, β ∈ Aut(R ⊂ M0). We view Aut(R ⊂M0) and the groups Γ0,Γ1 as subgroups of Out(R). Assume that Γ1 isfree with respect to the group Aut(R ⊂ M0). The following normalizingrelation

βσgβ−1 = (Ad(xδ(g))σδ(g), for all g ∈ Γ1 ,

obtained from (8.14) and the freeness assumption force the automorphismβ to be inner.

Step 4. Conjugating α by a unitary of R, we may assume, after Step 3,that α(x) = x, for all x ∈ R. The result follows, by Proposition 2.21.

Theorem 8.15 (Theorem 8.7 in [21]). Let Γ0,Γ1 be countable groups anddenote by R the hyperfinite II1 factor.

Assumptions on the groups Γ0,Γ1.


• Γ0 and Γ1 are freely indecomposable,

• Z,Γ0 and Γ1 are two by two not isomorphic.

Assumptions on the actions.

• The action Γ0 y R is outer and the inclusion R ⊂ Ro Γ0 is rigid.

• The action Γ1 y R is outer such that the natural action σ : Γ0 ∗Γ1 y R is also outer.

Freeness assumption.

The images in Out(R) of the countable subgroups σ(Γ1) and Aut(R ⊂Ro Γ0) (countable by Proposition 6.17) are free in Out(R).


Char(Γ0 ∗ Γ1) → Aut(Ro (Γ0 ∗ Γ1)

): ω → αω

induces an isomorphism Char(Γ0 ∗ Γ1) ∼= Out(Ro (Γ0 ∗ Γ1)

).

Proof. We denote Mi := Ro Γi, i = 0, 1 and form the amalgamated freeproduct M := R o Γ = M0 ∗R M1. Let α be an automorphism of M .Since the map Char(Γ0 ∗Γ1) → Aut

(Ro (Γ0 ∗Γ1)

): ω → αω is obviously

injective, we are left with proving the existence of a character ω of thegroup Γ such that α = αω, modulo unitary conjugacy. This will followfrom the four steps described previously.

Step 1. Because of Lemma 8.10, and because α is an automorphism ofM , we only have to prove that

α(R) ≺MR .

Since M = R o Γ, α(R) ⊂ M is a regular subfactor. The inclusionα(R) ⊂M is rigid and Corollary 8.9 allows us to conclude.

Step 2. Proposition 2.19 yields an automorphism δ of Γ such that

α(ug) = xδ(g)uδ(g), for all g ∈ Γ . (8.15)

The assumptions on the groups Γ0,Γ1 and the Kurosh automorphismtheorem applied to δ yield an element g ∈ Γ and automorphisms δi ∈


Aut(Γi) such that δ = Ad(g)(δ0 ∗ δ1

). Conjugating α from the beginning

by the unitary ug, we may assume that δ = δ0 ∗ δ1.

Step 3. Let β := α|R. Step 1 implies that β(R) = R and by Step 2, wehave that β(M0) = M0. Then β ∈ Aut(R ⊂ M0). We compute, for allg ∈ Γ1 and x ∈ R

(βσgβ−1)(x) = β(ugβ−1(x)u∗g)

= (Adxδ(g))uδ(g)xu∗δ(g), by(8.15)

= (Adxδ(g))σδ(g)(x) .

So βσgβ−1 = σδ(g) in Out(R). Note that Step 2 implies in particular

that δ(Γ1) ⊂ Γ1. Because of freeness assumption, the automorphism β istrivial in Out(R). Then, conjugating from the beginning α by a unitaryin M , we have that α(x) = x, for all x ∈ R.

Take now Γ0 = SL(3,Z) and Γ1 = SL(3,Z) × Char(G), where G is anarbitrary second countable abelian compact group. It is proven in [21,Lemma 8.6] that these groups satisfy all requirements of Theorem 8.15.Since Γ0 only has trivial characters, Theorem 8.15 yields a II1 factor Msuch that Out(M) ∼= Char(Γ1) = G. So we have the following.

Corollary 8.16 (and Corollary 8.8 in [21]). Every abelian compact groupis the outer automorphism group of a II1 factor.

Chapter 9

Every compact group is the

outer automorphism group

of a II1 factor

This chapter is a detailed version of the article [15] which is a joint workwith Stefaan Vaes. The main result that we prove in this chapter isthat every compact second countable group can be realized as the outerautomorphism group of a type II1 factor. At the end of this chapter, wegive a (shorter) proof that differs from the one in [15] and has not beenpublished.

In Chapter 8 we gave Ioana, Peterson and Popa’s proof of the fact thatevery abelian compact group arises as the outer automorphism group of aII1 factor. This result was the motivation for the present work. We neededto generalize Ioana, Peterson and Popa’s methods in order to obtain alsoall non-abelian compact groups. Before giving a precise statement of ourresult, we outline the main steps of the proof, explaining our strategy.

Our factors are also amalgamated free product II1 factors of the formM = M0 ∗N M1, where N is a copy of the hyperfinite II1 factor. Ioana,Peterson and Popa’s factors are of the form M = N o (Γ0 ∗Γ1), such thatN ⊂M0 has relative property (T) (see section 2.3 for the notion of relativeProperty (T)). Writing M = M0 ∗N M1, with Mi = N o Γi, i = 0, 1 theyhad the following data.

• A rigid inclusion N ⊂M0,

172 Chapter 9. II1 factors M with prescribed compact Out(M).

• An irreducible and regular subfactor N ⊂M1.

We are going to study, more generally, quasi-regular and irreducible in-clusions N ⊂M and assume that N ⊂M0 has relative property (T) in avery strong sense: we suppose that M0 itself has property (T). Our modelfor such inclusions is the following. We consider a minimal action of acompact group σ : G y M1 on the hyperfinite II1 factor M1 and denoteN := MG

1 , the fixed point II1 factor. The leg M0 remains the same as inIoana, Peterson and Popa’s work: it is of the form M0 = N o Γ, where Γis a countable group acting outerly on N . Let α ∈ Aut(M).

Step 1. Up to unitary conjugacy, α ∈ Aut(N ⊂M0).

By Aut(N ⊂M0) we mean the automorphisms of M0 leaving N globallyinvariant. One of the key points in Ioana, Peterson and Popa’s proof isthat whenever α is an automorphism of M , then α(N) = N and α(M0) =M0, up to conjugating by a unitary in M . In our construction, the twolegs M0 and M1 have very different behavior: one has property (T) andthe other is hyperfinite. We exploit this difference and use Theorem 8.7and Theorem 8.8 to prove that α(M0) = M0, up to unitary conjugacy.Then, since M0 is a crossed product and since Corollary 8.9 holds forquasi-regular inclusions we can apply Ioana, Peterson and Popa’s proof(see step 1 in Theorem 8.15) to α|M0

and get, similarly, that α(N) = N ,up to unitary conjugacy in M .

Step 2. A Kurosh automorphism theorem for fusion algebras.

The factors M we consider here are, a priori, amalgamated free productII1 factors that are not of crossed product type so we do not have thenice formula (8.15) expressing α in terms of an automorphism of a group.But since M0 = N o Γ, we can decompose L2(M0) into a direct sumof irreducible N -N -bimodules indexed by Γ. Moreover, since G y M1

minimally, we have seen in Chapter 7 that L2(M1) is the direct sum ofthe spectral subspaces L(π) associated to every irreducible representationπ ∈ Irr(G). These two decompositions provide a decomposition of L2(M)in terms of N -N -bimodules H(ω), where ω belongs to the free monoidΓ ∗ Irr(G). Since Γ acts outerly on N and since G acts minimally on M1,Proposition 6.6 and Theorem 7.24 allow us to view N[Γ] and Rep(G) asfusion subalgebras of the fusion algebra FAlg(N). If we assume that N[Γ]and Rep(G) are free inside FAlg(N), the N -N -bimodules H(ω) followirreducible (see section 6.1.5 for the notion of freeness for fusion algebras).

173

We extend the automorphism α ∈ Aut(N ⊂ M0) into an automorphism∆α of the fusion algebra generated by the finite index N -N -bimodulesincluded in L2(M). So, by our freeness assumption, ∆α ∈ Aut(N[Γ] ∗Rep(G)). We prove a Kurosh type theorem for fusion algebras in Theorem9.8 allowing us to assume that ∆α = δ0 ∗δ1, where δ0 is an automorphismof Γ and δ1 an automorphism of Rep(G).

Step 3. α(M1) = M1.

The definition of the automorphism ∆α and the decomposition of L2(M)in terms of bimodules H(ω), ω ∈ Γ ∗ Irr(G) yield that

α(L0(π)) = L0(δ1(π)), for all π ∈ Irr(G) ,

and step 3 follows, since L0(π) generate M1.

Step 4. α(x) = x for all x ∈ N .

In this step, we make use of the freeness assumption. Since Γ y N outerlyand α ∈ Aut(N ⊂ M0), Proposition 2.21 yields a character ω ∈ Char(Γ)such that α|M0

= αω, once step 4. is proven (see notation 2.22). Similarly,since σ : G y M1 is minimal, Proposition 7.25 yields an element g ∈ G

such that α|M1= σg. Both automorphisms αω and σg agree on N and

thus α = αω ∗ σg. Our theorem is proven when all our assumptions canbe fulfilled for a countable group Γ with no non-trivial characters.

In the last section of this chapter, we give an alternative proof of ourmain theorem avoiding the use of a Kurosh automorphism theorem forfusion algebras. In this new proof, we make a better use of the freenessassumption and we are able to prove directly that α(M1) = M1.

Our main theorem in this chapter is the following.

Theorem 9.1. Let M1 be the hyperfinite II1 factor and G a compactgroup acting on M1. Denote N = MG

1 , the von Neumann algebra ofG-fixed points in M1. Let Γ be an ICC group acting on N . DenoteM0 := N o Γ. Assume that

1. the action σ : G y M1 is minimal,

2. the action Γ y N is outer and M0 has the property (T),

3. the natural images of RepG → FAlg(N) and Aut(N ⊂ M0)restr−→

Out(N) ⊂ FAlg(N) inside the fusion algebra FAlg(N), are free inthe sense of Definition 6.14.



Char(Γ)×G→ Aut(M0 ∗NM1) : (ω, g) 7→ αω ∗ σg

induces an isomorphism Char(Γ)×G ∼= Out(M0 ∗NM1).

Combining Theorem 6.15 and Theorem 9.1, we shall prove the following.

Corollary 9.2. Let G be a compact, second countable group and σ : G yR a minimal action on the hyperfinite II1 factor R. Let Γ := SL(3,Z).Then there exists an outer action of Γ on the fixed point algebra RG, suchthat for M given as the amalgamated free product M = (RG o Γ) ∗

RGR,

the natural homomorphism

G→ Aut(M) : g 7→ id ∗ σg

induces an isomorphism G ∼= Out(M).

So, we immediately get the following result.

Theorem 9.3. Let G be a compact, second countable group. There existsa type II1 factor M with Out(M) ∼= G.

9.1 Notations and preliminary lemmas

Minimal action G y M1.

Let σ be a minimal action of a second countable compact G on the II1 fac-tor M1 and denote N := MG

1 the fixed-point algebra. We fix a set Irr(G)of inequivalent and irreducible unitary representations of G. We recall acertain number of facts and notations concerning spectral subspaces ofirreducible representations of G and refer to Chapter 7 for proofs.

For every π ∈ Irr(G), we choose a unitary Vπ ∈ B(Hπ) ⊗M1 satisfying(id⊗ σg)(Vπ) = Vπ(π(g)⊗ 1); see Proposition 7.13. With these unitariesVπ we can build finite index irreducible bimodules, as follows (see section7.3). For every π ∈ Irr(G), the map

ψπ : N → B(Hπ)⊗N : x 7→ Vπ(1⊗ x)V ∗π

9.1 Notations and preliminary lemmas 175

is an irreducible inclusion yielding the following irreducible finite indexN -N -bimodule

H(ψπ) := N

(H∗π ⊗ L2(N)

)ψπ(N) .

Define L0(π) ⊂M1 as the linear span of

(ξ∗ ⊗ a)Vπ(η ⊗ 1) , ξ, η ∈ Hπ , a ∈ N .

It follows that the closure L(π) of L0(π) is a finite index N -N -subbimodu-le of L2(M1).

Because of Corollary 7.23 we can decompose L2(M1) as the followingdirect sum of N -N -bimodules

L2(M1) =⊕

π∈Irr(G)

L(π) (9.1)

∼=⊕

π∈Irr(G)

(dimπ)H(ψπ) . (9.2)

Outer action Γ y N .

Let Γ be an ICC countable group acting outerly on the II1 factor Npreviously defined. We denote M0 := N o Γ. Define, for every s ∈ Γ, thesubspace H0(s) ⊂M0 given by

H0(s) = Nus .

It follows that the closureH(s) ofH0(s) is a finite indexN -N -subbimodu-le of L2(M0) and L2(M0) decomposes as the direct sum of finite index andirreducible N -N -bimodules

L2(M0) =⊕s∈Γ

H(s) . (9.3)

The II1 factor M := M0 ∗N M1.

Whenever w = s0π1s1 · · · sn−1πnsn is an alternating word in Γ \ e andIrr(G) \ e, we define the N -N -subbimodule H(w) ⊂ L2(M) as theclosure of

H0(s0)L0(π1) · · ·L0(πn)H0(sn) .


Because of the orthogonal decompositions (9.1) and (9.3), we immediatelyobtain the orthogonal decomposition of L2(M) given by

L2(M) =⊕

w alternating word

H(w) . (9.4)

Moreover, as N -N -bimodules, we get the unitary equivalences,

H(w) ∼= H(s0)⊗N L(π1)⊗N · · · ⊗N L(πn)⊗N H(sn)∼= k · H(s0)⊗N H(ψπ1)⊗N · · · ⊗N H(ψπn)⊗N H(sn) .

(9.5)

with k =∏ni=1 dimπi. The first equivalence of N -N -bimodules is ob-

tained using the freeness with amalgamation. A similar formula is provenin Lemma 10.9 in a more general setting so we do not give a proof of thisisomorphism here. See also Remark 10.8.

Lemma 9.4. Under the assumptions of Theorem 9.1, the inclusion N ⊂M is irreducible and quasi-regular.

Proof. In the decomposition (9.4), we remark that L2(M) contains onlyonce the trivial N -N -bimodule; for w the empty alternating word andH(w) = L2(N). This implies in particular that N ′ ∩M = C1. Indeed,if it was not the case, any non-trivial element x ∈ N ′ ∩M would yield anon-trivial N -N -bimodule Nx isomorphic to L2(N).

By minimality of the action G y M1, the inclusion N ⊂M1 follows quasi-regular, by Lemma 7.12. The factor N is normalized by the unitariesug, g ∈ Γ so N ⊂ M0 is regular. Since we have the obvious inclusionQNM0

(N)′′ ∗N QNM1(N)′′ ⊂ QNM (N)′′, we are done.

Remark 9.5. Note that since N ⊂M is irreducible, M is a factor.

Notation 9.6. Denote by F0 the fusion subalgebra of FAlg(N) generatedby the finite index N -N -subbimodules of NL2(M)N.

Under the freeness assumption of Theorem 9.1, the N -N -bimodules H(w)previously defined are two by two disjoint and we can describe the fusionalgebra F0 as follows.

Lemma 9.7. Under the assumptions of Theorem 9.1, the map

Ψ : w 7→ H(s0)⊗N H(ψπ1)⊗N · · · ⊗N H(ψπn)⊗N H(sn) ,

where w = s0π1s1 · · · sn−1πnsn is an alternating word in Γ \ e andIrr(G)\e, extends to a fusion algebra isomorphism N[Γ]∗Rep(G) ∼= F0.

9.2 A Kurosh automorphism theorem for fusion algebras 177

Proof. The map Ψ is well defined, by (9.5). It is surjective by (9.4). Sup-pose now that for w = s0π1s1 · · · sn−1πnsn and w′ = s′0π

′1s′1 · · · s′m−1π

′ns′m

alternating words in Γ \ e and Irr(G) \ e we have

H(s0)⊗N H(ψπ1)⊗N · · · ⊗N H(ψπn)⊗N H(sn)∼= H(s′0)⊗N H(ψπ′1)⊗N · · · ⊗N H(ψπ′m)⊗N H(s′m) .

The freeness condition implies in particular that Γ and Rep(G) are freeinside FAlg(N) and thus n = m, H(si) ∼= H(s′i) and H(ψπi) ∼= H(ψπ′i),for all i = 0, . . . , n. Outerness of Γ y N yields si = s′i, for all i = 0, . . . , n.We also have that πi = π′i, for all i = 0, . . . , n, using Theorem 7.24 andthus w = w′, proving injectivity of the map Ψ.

Note that in this proof we used a weaker freeness assumption than theone in Theorem 9.1: we only assume that Γ and Rep(G) are free insideFAlg(N).

9.2 A Kurosh automorphism theorem for fusion

algebras

An important ingredient in the analysis of all automorphisms of an amal-gamated free product M = M0 ∗N M1 as in Theorem 9.1 above, is ageneralization of the Kurosh automorphism theorem to automorphismsof free products of fusion algebras. We do not prove a general result, buta rather easy theorem sufficient for our purposes.

Recall that a group is said to be freely indecomposable if it cannot bewritten as a non-trivial free product.

Theorem 9.8. Let Γ be a countable group non isomorphic to Z and freelyindecomposable. Let A be an abelian fusion algebra, with a dimensionfunction d, non-isomorphic to the group Z and non-isomorphic to Γ.

Every dimension preserving automorphism α of N[Γ] ∗ A is of the form(Adu) (α0 ∗ α1) for some u ∈ Γ ∗ grp(A) , α0 ∈ Aut(Γ) and α1 adimension preserving automorphism of A .

Proof. Let α be a dimension preserving automorphism of N[Γ] ∗ A .


We denote Λ = grp(A) , the intrinsic group of A , and ∆ = Γ ∗Λ , whichis as well the intrinsic group of N[Γ] ∗ A . We also write G = IrredA ,

which means that A = N[G] . We may of course assume that G 6= Λ ,

because the group case of our theorem is covered by the classical Kuroshtheorem. Finally, we set G = G \ e and Γ = Γ \ e . If u ∈ ∆ , wewrite u−1 instead of u .

Claim. There exists x ∈ G \Λ and u ∈ ∆ such that α(x) ∈ u(G \Λ)u−1 .

Proof of the claim. Define λ = infd(x) | x ∈ G \ Λ ≥√

2 . Takex ∈ G \ Λ with d(x) <

√2λ . Write α(x) as an alternating word in G

and Γ . Suppose that in this expression of α(x) , there appears twice aletter from G \ Λ . Then the dimension of these two letters is greater orequal than λ ≥

√2 , making d(x) = d(α(x)) ≥

√2λ ; a contradiction.

So, we have shown that α(x) = uyv−1 with y ∈ G \ Λ and u, v ∈ ∆ .

We may assume that u, v are either equal to e , either end with a letterfrom Γ . Expressing the commutation of α(x) and α(x) , we find thatuyyu−1 = vyyv−1 . Since y 6∈ Λ , we find that yy 6= e and so u = v ,

proving the claim.

Observation 1. If x ∈ ∆ , y ∈ G and xy = yx , then x ∈ Λ . Thisfollows by analyzing reduced words in Γ and G .

Because of the claim and replacing α by (Adu−1) α , we may from nowon assume the existence of x, y ∈ G \Λ with α(x) = y . Whenever a ∈ Λ ,

α(a) belongs to ∆ and commutes with y . Observation 1 above impliesthat α(Λ) ⊂ Λ . Similarly, α−1(Λ) ⊂ Λ so that α(Λ) = Λ . It follows thatthe restriction of α to ∆ defines an automorphism of Γ ∗ Λ that globallypreserves Λ . The classical Kurosh theorem implies that α(Γ) = Γ .

Observation 2. If z ∈ G and α(z) ∈ ∆G∆ , then actually α(z) ∈ G .Indeed, write α(z) = urv−1 for r ∈ G and u, v ∈ ∆ either equal to e orwith their last letter in Γ . Writing out that α(z) = urv−1 and y = α(x)commute, it follows that u = v = e . A similar observation holds for α−1 .

It remains to prove that α(G) = G . Assume the contrary and define

δ = infd(z) | z ∈ G, ( α(z) 6∈ G or α−1(z) 6∈ G ) .

Take z ∈ G with d(z) <√

2δ such that α(z) 6∈ G or α−1(z) 6∈ G .Assume that we are in the case α(z) 6∈ G . By construction α(r), α−1(r) ∈G for every r ∈ G with d(r) < δ . Write α(z) as an alternating word inG and Γ . By observation 2, the expression for α(z) contains at least

9.3 Proof of Theorem 9.1 and Corollary 9.2 179

twice a letter from G \ Λ . Hence every letter in the expression for α(z)has dimension strictly smaller than δ . Applying α−1 and using the factthat α−1(Γ) = Γ , we have written z ∈ G as an alternating word in Γ

and G with more than 2 letters; a contradiction.

9.3 Proof of Theorem 9.1 and Corollary 9.2

A first step in the proof of Theorem 9.1 is the following lemma. Thecrucial ingredients of its proof are Theorems 8.7 and Theorem 8.8.

Lemma 9.9. Suppose that the assumptions of Theorem 9.1 are fulfilled.Set M = M0 ∗N M1. For every α ∈ Aut(M), there exists u ∈ U(M) suchthat

((Adu) α

)(M0

)= M0 and (Adu) α ∈ Aut(N ⊂M).

Note that in fact assumption 3 in Theorem 9.1 will not be used in theproof of this lemma.

Proof. By Theorem 8.8 and because M0 has property (T), there existsi ∈ 0, 1 such that α(M0) ≺M Mi. Since M1 is hyperfinite, it followsthat i = 0. So, by Theorem 5.19, we can take a projection p ∈ Mn

0 , anon-zero partial isometry v ∈ (Cn)∗ ⊗M and a unital ∗-homomorphismψ : M0 → pMn

0 p satisfying

α(a)v = vψ(a), for all a ∈M0 . (9.6)

Since ψ(M0) has property (T), we know that ψ(M0) 6≺M0 N . By Theorem8.7, it follows that ψ(M0)′ ∩ pMnp ⊂ pMn

0 p. In particular, v∗v ∈ pMn0 p.

So, we may assume that p = v∗v, by Remark 5.20. Since also α(M0)′ ∩M ∼= M ′

0 ∩ M = C1, we have vv∗ = 1. Factoriality of M0 allows usto assume that v is a unitary; see the proof of Lemma 8.10 where thisclassical argument is detailed.

Applying the same reasoning to α−1, we also get a unitary w ∈ U(M)satisfying w∗M0w ⊂ α(M0). It follows that (wv)∗M0(wv) ⊂ M0. Sincethe unitary wv quasi-normalizes M0 inside M , another application ofTheorem 8.7 implies that wv ∈M0. But then all the inclusions

(wv)∗M0(wv) ⊂ v∗α(M0)v ⊂M0


are equalities. So, we may assume that α(M0) = M0, after conjugatingα by a unitary in M .

We prove now that α(N) = N , up to unitary conjugacy. Since we haveproven that α is an automorphism of the crossed-product M0, we canidentically apply Ioana, Peterson and Popa’s proof. Indeed, since N ⊂M

is quasi-regular (see Lemma 9.4), Corollary 8.9 implies that α(N) ≺M0 N .Similarly N ≺M0 α(N). The lemma then follows by applying Lemma8.10.

Proof of Theorem 9.1. We still denote M = M0 ∗N M1. Let α ∈Aut(M). The proof of our theorem will be complete once we have proventhat after a unitary conjugacy of α, one has

α(a) = a, for all a ∈ N, α(M0) = M0 and α(M1) = M1 .

This then implies that α|M1 = σg for some g ∈ G, by Proposition 7.25.So we have that α|M0 = αω for some ω ∈ Char(Γ). Hence, the homomor-phism

Char(Γ)×G→ Out(M) : (ω, g) 7→ αω ∗ σgis surjective. The injectivity of this homomorphism follows from the ir-reducibility N ′ ∩M = C1, proven in Lemma 9.4. Indeed, suppose thatthere exists u ∈ U(M) such that αω ∗ σg = Adu. Since αω and σg aretrivial on N , the unitary u commutes to N and thus u = 1.

Recall that we denote by CharG ⊂ Irr(G) for the subset of Irr(G) con-sisting of one-dimensional unitary representations of G; see notation 2.4.Then, CharG is as well the intrinsic group of the fusion algebra RepG.Whenever π ∈ CharG, we have Vπ ∈ U(M1) and Vπ normalizesN . When-ever w ∈ Γ∗CharG, write w = s0π1s1 · · · sn−1πnsn as an alternating wordin Γ \ e and CharG \ e and define the unitary

U(w) := us0Vπ1 · · ·Vπnusn , (9.7)

normalizing N .

We are now ready to complete the proof of the theorem. So, let α ∈Aut(M). By Lemma 9.9, we may assume that α(N) = N . In orderto keep light notations, we view α as an automorphism of N , withoutwriting α|N . Then, the conjugation map

i(α) : F0 → F0 : K 7→ H(α−1)⊗N K ⊗N H(α)


defines an automorphism of the fusion subalgebra F0 of FAlg(N); seenotation 9.6 for F0. So we obtain the following automorphism η of N[Γ]∗Rep(G), where the isomorphism Ψ was defined in Lemma 9.7.

N[Γ] ∗ Rep(G)

η

""EEEEEEEEEEEEEEEEEEEEΨ // F0

i(α)

F0

Ψ−1

N[Γ] ∗ Rep(G)

So the automorphism η satisfies

Ψ(η(w)) = i(α)(Ψ(w)

), for all w ∈ N[Γ] ∗ Rep(G) . (9.8)

By Theorem 9.8, we find an element v in Γ∗Char(G) such that (Ad v)η =δ0 ∗ δ1, where δ0 is an automorphism of Γ and δ1 an automorphism ofRep(G). We have the following formula, where the element U(v) is theunitary defined by formula (9.7).

Ψ(v−1)⊗N i(α)(·)⊗N Ψ(v) ∼= i(α (AdU(v))

)(·) . (9.9)

We prove this formula and write v = s0π1s1 · · · sn−1πnsn, reduced wordin Γ ∗ CharG. Denoting by ρ the action Γ y N we have ρs = Adus, forall s ∈ Γ. Similarly, we have ψπ = AdVπ, for all π ∈ CharG. So

Ψ(v) = H(s0)⊗N H(ψπ1)⊗N · · · ⊗N H(ψπn)⊗N H(sn)∼= H(ρs0ψπ1 · · ·ψπnρsn)

= H(AdU(v))

Note that for every π ∈ CharG, there exists a unitary Xπ ∈ N such thatVπ−1 = Xπ(Vπ)∗. This formula and the fact that the unitaries us, s ∈ Γ,and Vπ, π ∈ CharG, normalize N inside M easily yield, by induction onthe length of words in Γ ∗CharG, the existence of a unitary X ∈ N suchthat

AdU(v−1) = (AdX) (AdU(v)∗) .


Then, we have, for every element K ∈ F0,

Ψ(v−1)⊗N i(α)(K)⊗N Ψ(v)

= Ψ(v−1)⊗N H(α−1)⊗N K ⊗N H(α)⊗N Ψ(v)∼= H

((AdX) (AdU(v)∗) α−1

)⊗N K ⊗N H

(α (AdU(v))

)∼= H(AdX)⊗N H

((AdU(v)∗) α−1

)⊗N K ⊗N H

(α (AdU(v))

)∼= L2(N)⊗N H

((AdU(v)∗) α−1

)⊗N K ⊗N H

(α (AdU(v))

)∼= i(α AdU(v)

)(K).

So, by the formula (9.9), we have, for every word ω ∈ N[Γ] ∗ Rep(G),

Ψ−1(i(α AdU(v−1))Ψ(ω)

)= Ψ−1

(Ψ(v)⊗N i(α)(Ψ(ω))⊗N Ψ(v−1)

)= Ad v η(ω)

= δ0 ∗ δ1 .

Since the unitary U(v) normalizes N , the automorphism α (AdU(v−1))still preserves N . Replacing α by α (AdU(v−1)), we may assume thatη = δ0∗δ1. Note that our Kurosh Theorem 9.8 yields a dimension preserv-ing automorphism of fusion algebras so the automorphism δ1 preservesthe fusion rules. Moreover, we have for every π ∈ Irr(G)

i(α)(L(π)) ∼= i(α)(dimπ H(ψπ)

)∼= dimπ Ψ(δ1(π)), by formula (9.8)∼= L(δ1(π)) .

Choose π ∈ Irr(G). The bimodule i(α)(L(π)) is isomorphic as an N -N -bimodule with the closure of α−1(L0(π)) in N L2(M)N ; we explain thisfact. Because of Remark 6.16, it is enough to prove that α(N)L(π)α(N)

∼=[α−1(L0(π))]. The obvious map a 7→ α−1(a) yields the required isomor-phism of N -N -bimodules.

Since i(α)(L(π)) = L(δ1(π)), the N -N -bimodules H(ψπ) and H(ψδ1(π))appear with the same multiplicity dimπ in the decomposition (9.4) ofL2(M). We conclude that the closure of α−1(L0(π)) inside L2(M) equalsL(δ1(π)) for all π ∈ Irr(G). It follows that α−1(L0(π)) = L0(δ1(π)) andthus α(M1) = M1.

Since α ∈ Aut(N ⊂ M0), assumption 3 in Theorem 9.1 implies that theN -N -bimodule H(α|N ) is free with respect to Ψ(RepG) inside FAlg(N).


But the formula i(α)(L(π)) ∼= L(δ1(π)) means that H(α|N ) normalizesΨ(RepG). Both statements can only be true at the same time if H(α|N )is the trivial N -N -module. So, α|N is an inner automorphism of N andwe are done.

Proof of Corollary 9.2. We claim that it suffices to give an exampleof an outer action of Γ = SL(3,Z) on the hyperfinite II1 factor N suchthat N o Γ has property (T).

Indeed, we start from a minimal action of G on the hyperfinite II1 factorM1 = R, set N = MG

1 . We take an outer action ρ : Γ y N such thatM0 := N o Γ has property (T). Denote by F the image in Out(N) of thegroup Aut(N ⊂ N o Γ). By Proposition 6.17, this group is countable.Since N is also hyperfinite, we have an isomorphism α : N → R. ByTheorem 6.15, we can choose α in a Gδ dense subset of Aut(R) such thatthe fusion algebras αFα−1 and Rep(G) are free inside FAlg(N). Then,replacing from the beginning the action ρ of Γ on N by αρα−1, all theconditions of Theorem 9.1 are fulfilled. Since Char Γ = e, the corollarythen follows from Theorem 9.1.

Take Γ1 := SL(3,Z) n (Z3 ⊕ Z3) with the action of SL(3,Z) on Z3 ⊕ Z3

given

A · (x, y) := (Ax, (A−1)ty) .

Note that Γ1 is a property (T) group. Take k ∈ R \ 2πQ and define thenon degenerate 2-cocycle Ω ∈ Z2(Z3 ⊕ Z3, S1) by the formula

Ω((x, y); (x′, y′)

):= eik(〈x,y

′〉−〈y,x′〉)

where 〈·, ·〉 is the standard scalar product on Z3; see section 2.2. The2-cocycle Ω is SL(3,Z)-invariant and hence extends to a 2-cocycle Ω ∈Z2(Γ1, S

1)

Ω(((x, y), A); ((x′, y′), B)

):= Ω

((x, y);A · (x′, y′)

),

by Lemma 2.3. The twisted group von Neumann algebra LΩ(Γ1) stillhas property (T) and can be regarded as well as LΩ(Z3 ⊕Z3) o SL(3,Z).Because of Proposition 2.5, LΩ(Z3 ⊕ Z3) is the hyperfinite II1 factor andwe are done.


9.4 Alternative proof of the main theorem

Here we present a shorter proof of Theorem 9.1. Making a better useof the freeness assumption, we are able to avoid the use of our Kuroshtheorem and prove directly that α(x) = x, for all x ∈ N and α(M1) = M1.

All the arguments in the beginning remain the same so we have an auto-morphism α of M = M0 ∗N M1 such that α(N) = N and α(M0) = M0.As noticed previously, since α(N) = N , the conjugation by α|N yields anautomorphism of F0. So

H(α−1|N )⊗N F0 ⊗N H(α|N ) = F0 . (9.10)

In Lemma 9.7, we have proven that F0 = N[Γ]∗Rep(G). Denote by F theimage in Out(N) of the group Aut(N ⊂ M0) and take π ∈ Irr(G) \ e.Note that H(α|N ) ∈ F . Then,

H(α−1|N )⊗N H(ψπ)⊗N H(α|N ) ∈ N[Γ] ∗ Rep(G) ⊂ F ∗ Rep(G) . (9.11)

Since we assumed that F and Rep(G) are free inside FAlg(N), the groupΓ is also free with Rep(G) and thus, formula (9.11) forces H(α|N ) ∈ N[Γ].Then, there exists an element g ∈ Γ such that, up to unitary conjugacyin N we have α = Adug. Conjugating from the beginning α by ug−1 , wehave that α(x) = x, for all x ∈ N .

We deduce now that α(M1) = M1. In the proof of Proposition 7.25 weobtained that

(id⊗ α)(Vπ) ∈ Vπ(B(Hπ)⊗ 1) . (9.12)

Since L0(π) is spanned by the set

(ξ∗ ⊗N)Vπ(η ⊗ 1) | ξ, η ∈ Hπ ,

formula (9.12) immediately implies that α(L0(π)) = L0(π) and thusα(M1) = M1.

Chapter 10

The representation category

of any compact group is the

bimodule category of a II1

factor

This chapter is a detailed version of the paper [16] that was written incollaboration with Stefaan Vaes. So far, we have given examples of II1factors for which one was able to compute the outer automorphism group.As already explained in Chapter 5, the computation of the fusion algebraor the entire bimodule category is a much more ambitious task. In [54],the scope of the methods of [21] was enlarged so that in certain cases notonly Out(M) but also Bimod(M) could actually be computed. The mainresult of [54] proves the existence of II1 factors M having trivial bimodulecategory and hence also trivial subfactor structure, trivial fundamentalgroup and trivial outer automorphism group. Note however that theresults in [21, 54] and in the previous chapter are existence theorems.The first concrete II1 factors with trivial bimodule category were givenin [53], which included as well concrete examples of II1 factors whereBimod(P ) is a Hecke-like category.

The main result we prove here is that the representation category of an ar-bitrary compact group G can be realized as the bimodule category of a II1factor. More precisely, we prove the existence of a minimal action G y M

186 Chapter 10. II1 factors with prescribed bimodule category.

of G on a II1 factor M , such that the bimodule category Bimod(MG) ofthe fixed point algebra MG is naturally isomorphic with the representa-tion category Rep(G). We have seen that, whenever G y M is a min-imal action, there is a natural embedding of Rep(G) into Bimod(MG),see Theorem 7.24. The striking point is that there exist minimal actionssuch that this embedding is surjective (up to unitary equivalence). Asin [21, 54] and the previous chapter, our result is an existence theorem,involving a Baire category argument (Theorem 6.15).

As in [21] and in the previous chapter, the II1 factor M in the previousparagraph is of the form M = M0 ∗N M1 and the action G y M is suchthat G acts trivially on M0, leaves M1 globally invariant and satisfiesMG

1 = N . Our main theorem is the following.

Theorem 10.1. Let G be a second countable compact group. There existsa II1 factor M and a minimal action G y M such that, writing P := MG,every finite index P -P -bimodule is isomorphic with PMor(L2(M),Hπ)Pfor a uniquely determined finite dimensional unitary representation π :G→ U(Hπ).

More precisely, Rep(G) → Bimod(P ) : π 7→ PMor(L2(M),Hπ)P definesan equivalence of C∗-tensor categories.

Theorem 10.1 provides examples of II1 factors for which all finite indexbimodules over P = MG can be listed explicitly, labeled by the finitedimensional unitary representations of G. Since the category of finiteindex bimodules over P encodes in a certain way all finite index subfactorsof P , these can be explicitly listed as well. In particular, Jones’ invariant[24]

C(P ) := [P : P0] | P0 ⊂ P irreducible, finite index subfactor

can be explicitly computed for the II1 factors P = MG given by Theorem10.1. The precise result goes as follows. We make use of Jones’ tunnelconstruction (see Theorem 4.16) saying that for every finite index inclu-sion of II1 factors P ⊂ N , there exists a finite index subfactor P0 ⊂ P

such that P0 ⊂ P ⊂ N is the basic construction. Moreover, P0 is uniquelydetermined up to unitary conjugacy in P .

Theorem 10.2. Let Gσy M be a minimal action of the second countable

compact group G on the II1 factor M and write P = MG. Assume that σ


satisfies the conclusion Theorem 10.1, meaning that every finite index P -P -bimodule is of the form PMor(L2(M),Hπ)P for some finite dimensionalunitary representation π of G.

Whenever Gαy A is an action on the finite dimensional von Neumann

algebra A with Z(A)G = C1, define the finite index subfactor P (α) ⊂ P

such that 1 ⊗ P (α) ⊂ 1 ⊗ P ⊂ (A ⊗M)α⊗σ is the basic construction.Here (α ⊗ σ)g := αg ⊗ σg and we note that P (α) is uniquely defined upto unitary conjugacy in P .

• Every finite index subfactor of P is unitarily conjugate to one of theP (α).

• [P : P (α)] = dimA and P (α) ⊂ P is irreducible iff AG = C1.

• If Gαy A and G

βy B satisfy Z(A)G = C1 and Z(B)G = 1, then

the subfactors P (α) and P (β) of P are unitarily conjugate in P iffthere exists a ∗-isomorphism π : A → B satisfying βg π = π αgfor all g ∈ G.

In particular, the set of index values of irreducible finite index subfactorsof P is given by

C(P ) = dim(A) | A finite dimensional von Neumann algebra,

G y A , AG = C1 .

10.1 Minimal actions and bimodule categories

We briefly recall some notations and results concerning minimal actionsof compact groups on II1 factors and refer to Chapter 7 for the proofsand further details.

Every minimal action G y M gives rise to a tensor functor

Rep(G) → Bimod(MG) : π 7→ PMor(L2(M),Hπ)P ,

see Theorem 7.24.

Let σ : G y M be a minimal action and choose a complete set Irr(G)of inequivalent, irreducible unitary representations of G. For every π ∈


Irr(G), we choose and fix a unitary Vπ ∈ B(Hπ) ⊗ M satisfying (id ⊗σg)(Vπ) = Vπ(π(g)⊗ 1); see Proposition 7.13.

Put P := MG. For every π ∈ Irr(G), the map

ψπ : P → B(Hπ)⊗ P : ψπ(a) = Vπ(1⊗ a)V ∗π

defines an irreducible, finite index inclusion, see Lemma 7.18. Definethe Hilbert space H(ψπ) = H∗

π ⊗ L2(P ), which is a P -P -bimodule asP(H∗

π ⊗ L2(P ))ψπ(P ). In Lemma 7.20, we proved that Mor(Hπ,L2(M)) ∼=H(ψπ) as P -P -bimodules.

We recall now some notations concerning spectral subspaces of irreduciblerepresentations introduced in section 7.2. Denote by Mor(Hπ,M) thespace of linear maps S : Hπ → M satisfying σg S = S π(g). Wedenote the linear span of Mor(Hπ,M)Hπ as L0(π) ⊂ M . The closure ofL0(π) inside L2(M) is denoted by L(π). In Proposition 7.22 we provedthe following.

• As a P -P -bimodule, L(π) is the direct sum of dim(π) copies ofH(ψπ). More precisely, if we consider on B(Hπ) ⊗ P the scalarproduct given by Tr⊗τ , the map

θπ : 1⊗ P

(B(Hπ)⊗ P

)ψπ(P ) → PL

0(π)P

such that

θπ(a) = dim(π)1/2(Tr⊗id)(aVπ), for all a ∈ B(Hπ)⊗ P ,

is P -P -bimodular, bijective and extends to an isometry B(Hπ) ⊗L2(P ) → L2(M).

• The adjoint of θπ is given by Eπ := θ∗π satisfying

Eπ(b) = dim(π)1/2∫G(π(g)∗ ⊗ σg(b))V ∗

π dg (10.1)

for all b ∈M .

• Since every unitary representation of G splits as a direct sum ofirreducibles, we have ∑

π∈Irr(G)

E∗πEπ = 1 . (10.2)

Equivalently, L2(M) is the orthogonal direct sum of the subspacesL(π), π ∈ Irr(G).

10.2 Proof of Theorem 10.1 189

Remark 10.3. Recall from Corollary 7.12 that the inclusion MG ⊂ M

is quasi-regular.

10.2 Proof of Theorem 10.1

Fix a second countable compact group G and an action Gσy M1 on the

II1 factor M1. Denote N = MG1 and fix an inclusion N ⊂M0 into the II1

factor M0. We are interested in the II1 factor

M := M0 ∗N M1

and extend the action G y M1 to an action G y M by acting triviallyon M0.

Assumptions

1. Assumption on the action Gσy M1 : σ is minimal and M1 is

hyperfinite.

2. Assumptions on the inclusion N ⊂M0.

2.a) The inclusion N ⊂M0 is irreducible, i.e. N ′ ∩M0 = C1 and isquasi-regular (see Section 2.1).

2.b) A condition on absence of finite dimensional unitary repre-sentations (cf. Lemmas 6.8 and 10.5). Denote by F0 the fu-sion subalgebra of FAlg(N) generated by the finite index N -N -subbimodules of NL2(M0)N. Whenever M0KM0 is an irreduciblefinite index M0-M0-bimodule containing a non-zero element ofF0 as N -N -subbimodule, we have M0KM0

∼= M0L2(M0)M0.

3. Rigidity assumption: there exists N ⊂ N0 ⊂ M0 such that N0

has property (T) in the sense of Connes and Jones and such thatN0 ⊂M0 is quasi-regular. See section 2.3 for the notion of property(T) for II1 factors.

4. Relation between Gσy M1 and N ⊂ M0. Denote by F the

fusion subalgebra of FAlg(N) generated by the finite index N -N -bimodules that arise as N -N -subbimodule of a finite index M0-M0-bimodule. Then, F is free with respect to the canonical image of


Rep(G) in FAlg(N), given by the minimal action σ (see Theorem7.24 and recall that N = MG

1 ). See section 6.1.5 for the notion offreeness for fusion algebras.

Remark 10.4. If N ⊂M0 is an irreducible inclusion of II1 factors havingthe relative property (T) (so, in particular, ifN ⊂M0 satisfies assumption3), the fusion algebra F defined in assumption 4 is countable, by Lemma6.12.

We now clarify the slightly mysterious assumption 2.b. The fusion subal-gebra F0 was previously introduced in section 6.1.3. The following lemmamakes the link with absence of finite dimensional unitary representations,see also Lemma 6.8 and Remark 6.9 following it.

Lemma 10.5. Let Γ be a countable group and Λ < Γ an almost normalsubgroup. Let Ω ∈ Z2(Γ, S1) be a scalar 2-cocycle. Put N := LΩ(Λ) andM0 := LΩ(Γ). If the following conditions hold, the inclusion N ⊂ M0

satisfies assumption 2.b.

• For all finite index subgroups Λ0 < Λ, we have LΩ(Λ0)′ ∩ LΩ(Γ) =C1.

• The group Γ has no non-trivial finite dimensional unitary represen-tations.

Proof. Define the fusion subalgebra F0 of FAlg(N) as in assumption 2.b.We first claim that every NHN in F0 is of good form, by which we meanthat there exists n and

• a finite index inclusion γ : N → Nn,

• elements g1, . . . , gn ∈ Γ and a finite index subgroup Λ0 < Λ ∩⋂ni=1 giΛg

−1i

such thatNHN ∼= γ(N)(Cn ⊗ L2(N))N and ,

γ(uh) =n∑i=1

eii ⊗ u∗giuhugi for all h ∈ Λ0 . (10.3)

We now prove the following three statements.


1. If NHN is of good form, then the same holds for all its N -N -subbimodules.

2. If NHN and NH′N are both of good form, then H′ ⊗N H is again of

good form.

3. We can decompose L2(M0) as a direct sum of N -N -subbimodulesof good form.

It is obvious that being of good form is preserved by direct sums. Fur-thermore, because NL2(M0)N is isomorphic with its contragredient, F0

is the smallest set of finite index N -N -bimodules (up to isomorphism),containing the finite index N -N -subbimodules of L2(M0) and being sta-ble under direct sums, tensor products and subbimodules. Hence, oncestatements 1, 2 and 3 are proven, our claim is proven as well.

Proof of 1. Let γ, g1, . . . , gn ∈ Γ and Λ0 be given as above. Leth1, . . . , hn ∈ Λ and W :=

∑ni=1 eii ⊗ uhi

∈ Nn. Then, for all h ∈ Λ0,

W ∗γ(uh)W =n∑i=1

eii ⊗ u∗gihiuhugihi

,

so if we replace gi by gihi and γ by a 7→ W ∗γ(a)W formula (10.3)still holds. So, we may assume that whenever gj ∈ giΛ, then gj = gi.Let s1, . . . , sk be the distinct elements of Γ such that g1, . . . , gn =s1, . . . , sk. By construction, sj 6∈ siΛ whenever i 6= j. So, there ex-ists a partition 1, . . . , n =

⊔ki=1 Ii such that gj = si, for all j ∈ Ii.

Then, the family of projections

pi :=∑j∈Ii

ejj ∈ Mn(C)

sum to 1 and satisfy, for all h ∈ Λ0,

γ(uh) =k∑i=1

pi ⊗ u∗siuhusi . (10.4)

Put Z :=∑k

i=1 pi⊗usi , which is a unitary in Mn0 . It follows that γ(uh) =

Z∗(1⊗ uh)Z for all h ∈ Λ0 and hence,

γ(N)′ ∩Nn ⊂ Z∗((1⊗ LΩ(Λ0))′ ∩Mn

0

)Z ⊂ Z∗(Mn(C)⊗ 1)Z . (10.5)


Let now q ∈ γ(N)′ ∩Nn be a projection. We have to prove that the N -N -bimodule defined by a 7→ γ(a)q is again of good form; see Proposition5.22. Because of (10.4), it suffices to prove that q = r ⊗ 1, where r ∈Mn(C) and rpi = pir for all i = 1, . . . , k. Indeed, if such projectionr exists, we have sub-projections ri ≤ pi such that r =

∑ki=1 ri. Let

i1, . . . , il be the indices i such that ri 6= 0. Let n′ =∑l

j=1 Tr(rij ). Takea matrix U ∈ Mn,n′(C) such that r′j := U∗rijU is diagonal, for all j =1, . . . , l. Then, replacing γ by (U∗ ⊗ 1)γ(·)(U ⊗ 1), we obtain that

γ(uh)q =l∑

j=1

r′j ⊗ u∗sijuhusij

, for all h ∈ Λ0 .

By (10.5), q = Z∗(t⊗ 1)Z for some t ∈ Mn(C). Let i 6= j. Then,

(pi ⊗ 1)q(pj ⊗ 1) = pitpj ⊗ Ω(s−1i , sj)us−1

i sj

belongs to Mn(C)⊗N and to Mn(C)⊗Cus−1i sj

. Hence, (pi⊗1)q(pj⊗1) = 0.But then,

q =k∑i=1

(pi ⊗ 1)q(pi ⊗ 1) =k∑i=1

pitpi ⊗ 1 ,

concluding the proof of 1.

Proof of 2. Suppose that NHN and NH′N are of good form. Suppose that

γ : N → Nn defines NHN and satisfies (10.3) with respect to g1, . . . , gn,Λ0 < Λ ∩

⋂ni=1 giΛg

−1i . Suppose that ρ : N → Nm defines NH′

N andsatisfies (10.3) with respect to h1, . . . , hm and Λ1 < Λ ∩

⋂mj=1 hjΛh

−1j .

Then, h−1j hhj ∈ Λ0, for all h ∈ Λ1 ∩

⋂mj=1 hjΛ0h

−1j and we have

(id⊗ γ)ρ(uh) =m∑j=1

ejj ⊗ γ(uh−1j hhj

)

=m∑j=1

n∑i=1

ejj ⊗ eii ⊗ u∗hjgiuhuhjgi

.

So, the composition (id ⊗ γ)ρ satisfies (10.3) with respect to the groupelements hjgi and the subgroup Λ1∩

⋂mj=1 hjΛ0h

−1j . It follows that H′⊗N

H is of good form, by Proposition 5.8.

Proof of 3. For every g ∈ Γ, define H(g) as the closure of NugN inL2(M0). Write

ΛgΛ =n⊔i=1

giΛ .


The map ei ⊗ a → ugia extends to a unitary V : Cn ⊗ L2(N) → H(g)satisfying V (ξa) = V (ξ)a for all ξ ∈ Cn ⊗ L2(N) and a ∈ N . Define themap γ such that V (γ(a)ξ) = aV (ξ) for all ξ ∈ Cn ⊗ L2(N) and a ∈ N .Arguing as in the proof of Proposition 5.1, the right N -linearity of Vimplies that γ commutes with the right N -action on Cn⊗L2(N). Hence,γ : N → Nn and

NH(g)N ∼= γ(N)

(Cn ⊗ L2(N)

)N .

The map γ satisfies (10.3) with respect to g1, . . . , gn and the subgroupΛ0 = Λ ∩

⋂ni=1 giΛg

−1i . Indeed, let h ∈ Λ0 and take ki ∈ Λ such that

hgi = giki, for all i = 1, . . . , n. Then, for all a ∈ N ,

γ(uh)(ei ⊗ a) = V ∗(uhugia)

= V ∗(ugiukia)

= ei ⊗ ukia

= ei ⊗ u∗giuhugi

=( n∑j=1

ejj ⊗ u∗gjuhugj

)(ei ⊗ a) .

Since the H(g), g ∈ Γ, are two by two orthogonal and their linear spanis dense in L2(M0), the final statement 3 is proven and hence, also ourclaim.

Fix now an irreducible finite index M0-M0-bimodule M0KM0 and let H ⊂K be an irreducible N -N -subbimodule such that NHN belongs to F0. Forevery g, h ∈ Γ, the closure of Nug · H · uhN inside K is isomorphic withH(g) ⊗N H ⊗N H(h) as N -N -bimodules. Hence, this closure belongs toF0 and the irreducibility of K implies that K is a direct sum of N -N -subbimodules that all belong to F0.

By Proposition 5.1, we can write M0KM0∼= ψ(M0)p(Cm ⊗ L2(M0))M0 for

some finite index irreducible inclusion ψ : M0 → pMm0 p. Define A :=

pMm0 p ∩ ψ(N)′. Since K is of finite index and N ⊂ M0 is irreducible,

it follows that A is finite dimensional, by Lemma 4.12. Let q ∈ A be aminimal projection. Because of the previous paragraph, we find a finiteindex inclusion γ : N → Nn satisfying (10.3) with respect to g1, . . . , gnand Λ0 and a bimodular isometry

θ : γ(N)(Cn ⊗ L2(N))N → ψ(N)q

(q(Cm ⊗ L2(M0))

)N .


Define ξ ∈ q(Cm(Cn)∗ ⊗ L2(M0)) by the formula

ξ :=n∑i=1

θ(ei ⊗ 1)(e∗i ⊗ 1) .

It follows that ξ is non-zero and satisfies ψ(a)ξ = ξγ(a) for all a ∈ N . Asan element of q(Mm(C) ⊗ L1(M0))q, the operator ξξ∗ commutes withψ(N). Hence, ξξ∗ is a multiple of q and we may assume that ξ ∈q(Cm(Cn)∗⊗M0) with ξξ∗ = q; see section 2.5. Define V = ξ(

∑i eii⊗u∗gi

).It follows that V ∈ q(Cm(Cn)∗⊗M0), V V ∗ = q and ψ(uh)V = V (1⊗uh)for all h ∈ Λ0. Since LΩ(Λ0) has trivial relative commutant in M0, it fol-lows that V ∗V = p1⊗1 for some projection p1 ∈ Mn(C). Let k := dim(p1)and take a matrix U ∈ Cn(Ck)∗ such that U∗p1U = 1. Hence, the elementW ∈ q(Cm(Ck)∗ ⊗M0) defined by W := V (U ⊗ 1) satisfies WW ∗ = q,W ∗W = 1 and ψ(uh)W = W (1⊗ uh) for all h in a finite index subgroupof Λ.

Le p1, . . . , pr be minimal orthogonal projections in A summing to 1. Theprevious paragraph yields elements Wi ∈ pi(Cm(Cni)∗ ⊗M0) and finiteindex subgroups Λi < Λ such that

• WiW∗i = pi ,

• W ∗i Wi = 1 ,

• ψ(uh)Wi = Wi(1⊗ uh) for all h ∈ Λi .

Set Λ0 :=⋂ri=1 Λi and

X := (W1, . . . ,Wr) ∈ p((Cm(Cn)∗⊗M0)

), where p :=

p1

. . .pr

.

Then, since the pi are orthogonal and sum to 1 it easy to check that Xis a unitary and satisfies X∗ψ(uh)X = 1⊗ uh for all h ∈ Λ0. So, we mayactually assume that ψ(uh) = 1⊗ uh for all h ∈ Λ0.

If now g ∈ Γ, we get that ψ(ug)(1 ⊗ u∗g) commutes with 1 ⊗ uh for allh ∈ Λ0∩gΛ0g

−1. Indeed, we have that g−1hg ∈ Λ0, for all h ∈ Λ0∩gΛ0g−1

so,

ψ(ug)(1⊗ u∗g)(1⊗ uh) = ψ(ug)(1⊗ ug−1hg)(1⊗ u∗g)

= ψ(ug)ψ(ug−1hg)(1⊗ u∗g)

= (1⊗ uh)ψ(ug)(1⊗ u∗g) .


By assumption, LΩ(Λ0 ∩ gΛ0g−1) has trivial relative commutant in M0

so ψ(ug) = π(g) ⊗ ug for all g ∈ Γ, where π : Γ → U(Cr) is a finitedimensional unitary representation. By assumption, π(g) = 1 for allg ∈ Γ. So, ψ(a) = 1 ⊗ a for all a ∈ M0. By irreducibility of K, we getr = 1 and M0KM0

∼= M0L2(M0)M0.

In fact, we only use the following concrete example satisfying the condi-tions of Lemma 10.5 and hence providing an inclusion N ⊂M0 satisfyingassumption 2, with N being isomorphic with the hyperfinite II1 factor.Moreover N ⊂M0 satisfies assumption 3.

Example 10.6. Consider the group Γ = (Q3 ⊕ Q3) o SL(3,Q), definedby the action A · (x, y) = (Ax, (At)−1y) of SL(3,Q) on Q3 ⊕Q3. Choosean irrational number α ∈ R and define, as in Lemma 2.3, the 2-cocycleΩ ∈ Z2(Γ, S1) such that

Ω((x, y), (x′, y′)

)= exp

(iα(〈x, y′〉 − 〈y, x′〉)

), for all (x, y),

(x′, y′) ∈ Q3 ⊕Q3 ,

Ω(g,A) = Ω(A, g) = Ω(A,B) = 1, for all g ∈ Γ ,

A,B ∈ SL(3,Q) .

Set Λ = Z3 ⊕ Z3. We define N := LΩ(Λ) and M0 := LΩ(Γ). We provethat N ⊂M0 satisfies assumptions 2 and 3.

Assumption 3 follows by taking N0 := LΩ

((Z3 ⊕ Z3) o SL(3,Z)

), which

has property (T) because (Z3 ⊕ Z3) o SL(3,Z) is a property (T) group;see section 2.3.

Since SL(3,Q) has no non-trivial finite-dimensional unitary representa-tions (see for example [59]) and since the smallest normal subgroup of Γcontaining SL(3,Q) is the whole of Γ, it follows that Γ has no non-trivialfinite-dimensional unitary representations. Because of Lemma 10.5, itremains to prove that for every finite index subgroup Λ0 < Λ, we haveLΩ(Γ) ∩ LΩ(Λ0)′ = C1. Write Λ1 = Q3 ⊕Q3. Take a ∈ LΩ(Γ) ∩ LΩ(Λ0)′

and write, with L2-convergence, a =∑

g∈Γ agug. We prove that ag = 0,for all g 6= e.

Step 1. ag = 0 for all g ∈ Γ \ Λ1.

An immediate computation yields

agh−1Ω(gh−1, h) = ah−1gΩ(h, h−1g), for all g ∈ Γ, h ∈ Λ0 . (10.6)


So we have |agh−1 | = |ah−1g|, for all g ∈ Γ and h ∈ Λ0. The set

Ig := hgh−1 | h ∈ Λ0

is infinite for all g ∈ Γ−Λ1 so we get ag = 0 for all g ∈ Γ−Λ1. To provethat Ig is infinite for all g ∈ Γ− Λ1, write g := (x,A), with x ∈ Q3 ⊕Q3

and A a non-trivial element of SL(3,Q). Then, for every y ∈ Λ0, weobtain

(y, 1)g(−y, 1) = (x+ (1−A)y,A) ,

from which it follows that Ig is infinite, since A is non-trivial.

Step 2. ag = 0 for all g ∈ Λ1 \ 0.Because of (10.6) we have

agΩ(g, h) = ah−1ghΩ(h, h−1gh), for all g ∈ Γ, h ∈ Λ0 .

In particular,

agΩ(g, h) = agΩ(h, g), for all g ∈ Λ1, h ∈ Λ0 . (10.7)

It follows from the definition of the bi-character Ω that Ω(g, h) = Ω(h, g).So, by (10.7), the map

π : Λ1 → Λ1 : h 7→ πh where πh(g) = Ω(h, g)−2

satisfies agπg(h) = ag for all g ∈ Λ1 and all h ∈ Λ0. By definition of Ω,if g ∈ Λ1 − 0, the character πg is not identically 1 on Λ0 and hence,ag = 0.

We deduce Theorem 10.1 from the following general statement.

Theorem 10.7. Under the assumptions at the beginning of the sectionand writing M = M0 ∗N M1, the action G y M is minimal and thenatural tensor functor defined in Theorem 7.24

Rep(G) → Bimod(MG) : π 7→ MGMor(L2(M),Hπ)MG

is an equivalence of categories.

The rest of this section is devoted to a proof of Theorem 10.7 and de-ducing 10.1 as a corollary. Denote P = MG and make throughout theassumptions made at the beginning of the section.


Choose a complete set Irr(G) of representatives for the set of irreducibleunitary representations of G modulo unitary conjugacy. Since σ : G yM1 is minimal, choose, for every π ∈ Irr(G), a unitary Vπ ∈ B(Hπ)⊗M1

satisfying (id ⊗ σg)(Vπ) = Vπ(π(g) ⊗ 1) for all g ∈ G. Define the finiteindex inclusions

ψπ : P → B(Hπ)⊗ P : ψπ(a) = Vπ(1⊗ a)V ∗π for all a ∈ P

and note that ψπ(N) ⊂ B(Hπ)⊗N . As before, we denote by H(ψπ) theN -N -bimodule given by N(H∗

π ⊗ L2(N))ψπ(N).

Remark 10.8. In Corollary 7.23, we proved that that the N -N -bimoduleNL2(M1)N can be decomposed into a direct sum of N -N -bimodules L(π),π ∈ Irr(G), where L(π) is isomorphic with dim(π) copies of H(ψπ). Fur-thermore L(π) is the closure of L0(π) ⊂ M1, in such a way that thelinear span of all L0(π) is an ultraweakly dense ∗-subalgebra of M1; seeProposition 7.6.

Since N ⊂M0 is irreducible and quasi-regular, we can decompose L2(M0)L2(N) as the orthogonal direct sum of irreducible, non-trivial, finiteindex N -N -subbimodules Ri, i ∈ I. Put R0

i := Ri∩M0. By Lemma 5.17,R0i is dense in Ri. By construction, R0

i ⊂ M0 N . Assume 0 6∈ I andput R0

0 := N . By Lemma 6.4, spanR0i | i ∈ I ∪ 0 equals QNM0

(N)and is, in particular, an ultraweakly dense ∗-subalgebra of M0.

Whenever n ∈ N ∪ 0, i0, in ∈ I ∪ 0, i1, . . . , in−1 ∈ I and π1, . . . , πn ∈Irr(G) \ ε, denote by R(i0, π1, . . . , πn, in) the closure of

R(i0, π1, . . . , πn, in)0 := R0i0L

0(π1)R0i1 · · ·R

0in−1

L0(πn)R0in

inside L2(M).

• The definition of the amalgamated free product implies that L2(M)is the orthogonal direct sum of the subspaces R(i0, π1, . . . , πn, in).

Furthermore, the freeness with amalgamation implies that the map

a0 ⊗ b1 ⊗ a1 ⊗ · · · ⊗ bn ⊗ an 7→ a0b1a1 · · · bnan

extends to a unitary from Ri0 ⊗N L(π1) ⊗N · · · ⊗N L(πn) ⊗N Rin ontoR(i0, π1, . . . , πn, in) (proven in Lemma 10.9).


• R(i0, π1, . . . , πn, in) is isomorphic with dim(π1) · · ·dim(πn) copies of

Ri0 ⊗N H(ψπ1)⊗N · · · ⊗N H(ψπn)⊗N Rin . (10.8)

Finally, the linear span of all R(i0, π1, . . . , πn, in)0 is an ultraweakly dense∗-subalgebra of M .

Denote by F0 the fusion subalgebra of FAlg(N) generated by the finiteindex N -N -subbimodules Ri, i ∈ I. Assumption 4 implies in particularthat the fusion subalgebra of FAlg(N) generated by H(ψπ), π ∈ Irr(G), isfree w.r.t. F0. Therefore the N -N -bimodules appearing in (10.8) are irre-ducible and we have found a decomposition of NL2(M)N as a direct sumof irreducible finite index N -N -bimodules. The trivial N -N -bimoduleappears with multiplicity one in L2(M). This means that N ′ ∩M = C1.In particular, the action G y M is minimal.

The action of G on L2(M) leaves every R(i0, π1, . . . , πn, in) globally in-variant and we denote by R(i0, π1, . . . , πn, in)G the subspace of G-fixedvectors. It follows that L2(P ) is the orthogonal direct sum of the N -N -subbimodules R(i0, π1, . . . , πn, in)G, which are, as N -N -bimodules iso-morphic with a multiple of the N -N -bimodule given by (10.8). Moreprecisely, we have the following.

In the forthcoming lemma and throughout the rest of this chapter, weuse the following notation. Let M1, . . . ,Mn be von Neumann algebras.Let a ∈Mn. We write

(a)n := 1⊗ 1⊗ · · · ⊗ 1︸︷︷︸n−1 times

⊗a ∈n⊗k=1

Mk .

Lemma 10.9. The von Neumann algebra P is (ultraweakly) denselyspanned by

(ξ∗ ⊗ 1)(a0)n+1(Vπ1)1,n+1 · · · (Vπn)n,n+1(an)n+1(η ⊗ 1)∣∣∣

ξ ∈n⊗i=1

Hπi , η ∈( n⊗i=1

Hπi

)G, (10.9)

where a0, . . . , an ∈ M0, π1, . . . , πn ∈ Irr(G) and where the superscriptG denotes the subspace of G-invariant vectors under the tensor product


representation. Moreover, L2(P ) decomposes as an orthogonal sum offinite index N -N -bimodules as follows.

L2(P ) =⊕

R(i0, π1, . . . , πn, in)G

∼=⊕

m(Ri0 ⊗N H(ψπ1)⊗N · · · ⊗N H(ψπn)⊗N Rin

),

where the integer m is given by the dimension of the space of (π1 ⊗ · · · ⊗πn)(G)-invariant vectors in Hπ1 ⊗ · · · ⊗Hπn.

Proof. We define the map(Hπ1 ⊗ . . .⊗Hπn

)⊗Ri0 ⊗N H(ψπ1)⊗N · · · ⊗N H(ψπn)⊗N Rin

Θ

R(i0, π1, . . . , πn, in) ,

such that, writing H(ψπ) = H∗π ⊗ L2(N), we have

Θ(η ⊗ a0 ⊗ (ξ∗1 ⊗ b1)⊗ . . .⊗ (ξ∗n ⊗ bn)⊗ an

)= (ξ∗1 ⊗ . . .⊗ ξ∗n ⊗ 1)(a0)n+1(b1)n+1(Vπ1)1,n+1 . . .

. . . (Vπn)n,n+1(an)n+1(bn)n+1(η ⊗ 1) ,

where ak ∈ R0ik

, ξk ∈ Hπk, bk ∈ N . We prove that Θ is a unitary satisfying

Θ (π1(g)⊗ . . .⊗ πn(g)⊗ 1

)= σg Θ, for all g ∈ G . (10.10)

Formula (10.10) is obvious from the definition of Θ and it implies that

R(i0, π1, . . . , πn, in)G ∼=(Hπ1 ⊗ . . .⊗Hπn

)G ⊗Ri0 ⊗N H(ψπ1)⊗N· · · ⊗N H(ψπn)⊗N Rin .

Identifying((R(i0, π1, . . . , πn, in)0

)G with the image of Θ, and using thefact that the R0

i span a dense ∗-subalgebra of M0 proves that P can bedensely spanned as in (10.9). So we are left with proving that Θ is aunitary. It is enough to prove that Θ is an isometry since it is denselydefined and has dense range, as we can see by re-writing Θ in the followingway.

Θ(η1 ⊗ · · · ⊗ ηn ⊗ (a0 ⊗ (ξ∗1 ⊗ b1)⊗ . . .⊗ (ξ∗n ⊗ bn)⊗ an)

)= a0

((ξ∗1 ⊗ b1)Vπ1(η1 ⊗ 1)

)a1 . . . an−1

((ξ∗n ⊗ bn)Vπn(ηn ⊗ 1)

)an ,


where ak ∈ R0ik

, ξk ∈ Hπk, bk ∈ N .

We compute the quantity

∆ := EN(a∗n(η

∗n ⊗ 1)V ∗

πnξnan−1 × . . .× a∗1(η

∗1 ⊗ 1)V ∗

π1ξ1(a∗0b0)µ

∗1Vπ1

(ν1 ⊗ 1)b1 × . . .× bn−1µ∗nVπn(νn ⊗ 1)bn

),

where ak, bk ∈ R0ik

, ηk, νk ∈ Hπk, ξk, µk ∈ Hπk

⊗ N . We start with thefollowing claim.

Claim 10.10. We define the following functions, for all b ∈M0.

φk(b) := EN (a∗kbbk) , k = 0, . . . , n ,

ψk(b) := (η∗k ⊗ 1)(id⊗ EN )(V ∗πkξkbµ

∗kVπk

)(νk ⊗ 1), k = 1, . . . , n.

Then, we have that

φk(b) = 〈ak, bbk〉N , ψk(b) = 〈ηk, νk〉〈ξ∗k, bµ∗k〉N .

Proof of the Claim. For the proof of the first formula, see Example 5.6.Furthermore,

ψk(b) = (η∗k ⊗ 1)(∫

G(id⊗ σg)

(V ∗πkξkbµ

∗kVπk

)dg

)(νk ⊗ 1)

= (η∗k ⊗ 1)(∫

G(πk(g)∗ ⊗ 1)V ∗

πkξkbµ

∗kVπk

(πk(g)⊗ 1) dg)

(νk ⊗ 1)

= (η∗k ⊗ 1)(tr⊗id)(V ∗πkξkbµ

∗kVπk

)(νk ⊗ 1)

= η∗kνk〈ξ∗k, bµ∗k〉N , by Lemma 7.19 ,

which proves the claim.

Before computing ∆, note, since R0i ⊂ M0 N that EN (ak) = 0, for all

k = 0, . . . , n. Moreover, since all πk are non-trivial we have that

EN((η∗k ⊗ 1)V ∗

πkξk)

= (η∗k ⊗ 1)(∫

G(πk(g)∗ ⊗ 1) dg

)Vπk

(νk ⊗ 1) = 0 .

Similarly, EN(µ∗kVπk

(νk ⊗ 1))

= 0. So we obtain

∆ = EN(. . . ξ1(a∗0b0 − EN (a∗0b0))µ

∗1 . . .

)+ EN

(. . . ξ1 EN (a∗0b0)µ

∗1 . . .

)= EN

(. . . ξ1 EN (a∗0b0)µ

∗1 . . .

), by freeness with amalgamation

= EN(. . . (η∗1 ⊗ 1)V ∗

π1ξ1φ0(1)µ∗1Vπ1(ν1 ⊗ 1) . . .

)= EN

(. . . a∗1ψ1 φ0(1)b1 . . .

), by freeness with amalgamation

...

= EN(a∗nψn φn−1 · · · ψ1 φ0(1)bn

)


On the other hand, we have

〈(η1⊗ . . .⊗ ηn)⊗ a0 ⊗ ξ∗1 ⊗ a1 ⊗ . . .⊗ ξ∗n ⊗ an , (ν1 ⊗ . . .⊗ νn)⊗ b0

⊗ µ∗1 ⊗ b1 ⊗ . . .⊗ µ∗n ⊗ bn〉

=n∏k=1

〈ηk, νk〉〈a0 ⊗ ξ∗1 ⊗ . . . , b0 ⊗ µ∗1 ⊗ . . . 〉

=n∏k=1

〈ηk, νk〉〈ξ∗1 ⊗ a1 ⊗ . . . , 〈a0, b0〉Nµ∗1 ⊗ b1 ⊗ . . . 〉

=n∏k=1

〈ηk, νk〉〈a1 ⊗ ξ∗2 ⊗ . . . , 〈ξ∗1 , φ0(1)µ∗1〉Nb1 ⊗ µ∗2 ⊗ . . . 〉

=n∏k=2

〈ηk, νk〉〈a1 ⊗ ξ∗2 ⊗ . . . , ψ1 φ0(1)b1 ⊗ µ∗2 ⊗ . . . 〉

...

=〈an , ψn φn−1 · · · ψ1 φ0(1)bn〉 .

We conclude by taking the trace of ∆.

Lemma 10.11. If M0HP is an irreducible non-zero M0-P -bimodule withdim(HP ) < ∞, there exists η ∈ Irr(G) and a non-zero M0-ψη(M0)-subbimodule K ⊂ H∗

η ⊗H with the following properties.

• dim(Kψη(M0)) <∞.

• If ρ ∈ Irr(G) and L ⊂ H∗ρ⊗H is a non-zero M0-ψρ(M0)-subbimodule

with dim(M0L) <∞, then ρ = η and L ⊂ K.

Proof. By Proposition 5.1, we can take ψ : M0 → pPnp such that

M0HP ∼= ψ(M0)p(Cn ⊗ L2(P ))P .

By assumption, the inclusion ψ(M0) ⊂ pPnp is irreducible. From as-sumption 3, we get the property (T) II1 factor N0 and hence, ψ(N0)has property (T) and is a subalgebra of pPnp ⊂ pMnp. Recall thatM = M0 ∗N M1 and that M1 is hyperfinite. Since there is no non-zerohomomorphism from a property (T) II1 factor to the hyperfinite II1 fac-tor, Lemma 8.13 provides an element u ∈ p(Cn(Ck)∗⊗M) with uu∗ = p,


q := u∗u ∈ Mk0 and u∗ QNpMnp

(ψ(N0)

)′′u ⊂ qMk

0 q. By quasi-regularityof N0 ⊂M0, we get

u∗ψ(M0)u = u∗ψ(QNM0

(N0)′′)u

⊂ u∗ QNpMnp

(ψ(N0)

)′′u ⊂ qMk

0 q .

Since uu∗ = p, we can define

γ : M0 → qMk0 q : a 7→ u∗ψ(a)u . (10.11)

We now use the bimodule maps Eπ given by (10.1). Take η ∈ Irr(G) suchthat (id⊗ Eη)(u) 6= 0. If such representation does not exist, we have∑

η

(id⊗ E∗ηEη)(u) = 0 ,

implying, by (10.2), that u = 0 and hence contradicting the fact that uis a unitary. So, we can take a vector ξη ∈ Hη such that

v := (1⊗ ξ∗η ⊗ 1)(id⊗ Eη)(u) ∈ p(Cn(Ck ⊗Hη)∗ ⊗ P )

is non-zero. Note that

(id⊗ Eη)(u) = dim(π)1/2(∫

G(1⊗ η(g)∗ ⊗ 1)

(id⊗ id⊗ σg)(u13) dg)

(1⊗ V ∗η ) (10.12)

and thus,

(id⊗ Eη)(u) ∈ p13

(Cn(Ck)∗ ⊗ B(Hη)⊗ P

)(id⊗ ψη)(q) .

Then, we have that

ψ(a)v = v(id⊗ ψη)γ(a), for all a ∈M0 . (10.13)

Indeed, for all a ∈M0,

ψ(a)v = (1⊗ ξ∗η ⊗ 1)(ψ(a)

)13

(id⊗ Eη)(u)

= (1⊗ ξ∗η ⊗ 1)(id⊗ Eη)(ψ(a)u)

= (1⊗ ξ∗η ⊗ 1)(id⊗ Eη)(uγ(a)), by (10.11)

= (1⊗ ξ∗η ⊗ 1) dim(π)1/2(∫

G(1⊗ η(g)∗ ⊗ 1)(1⊗ 1⊗ σg)

(u13) dg)(γ(a)

)13

(1⊗ V ∗η ), by G-invariance of γ(a)

= (1⊗ ξ∗η ⊗ 1)(id⊗ Eη)(u)(id⊗ ψη

)(γ(a)

).


Replacing v by its polar part, we may assume that v is a partial isometry.The irreducibility of ψ(M0) ⊂ pPnp ensures that vv∗ = p.

Define K as the closure of v(Ck ⊗ ψη(M0)). Because of (10.13), K isa ψ(M0)-ψη(M0)-subbimodule of p(CnH∗

η ⊗ L2(P )). By construction,dim(Kψη(M0)) <∞.

Let ρ ∈ Irr(G) and let L ⊂ p(CnH∗ρ ⊗ L2(P )) be a non-zero ψ(M0)-

ψρ(M0)-subbimodule with dim(ψ(M0)L) < ∞. We have to prove thatρ = η and L ⊂ K. Since dim(ψ(M0)L) < ∞, Proposition 5.1 yields aunital ∗-homomorphism ϕ : M0 → r

(ψ(M0)

)lr and an M0-M0-bimodule

isomorphism

T : ψ(M0)

((Cl)∗ ⊗ L2

(ψ(M0)

))r ϕ(M0) → ψ(M0)Lψρ(M0) .

Then, the remark following Proposition 5.11 yields a non-zero vector

ξ ∈ (1⊗ p)((Cl ⊗ Cn)H∗ρ ⊗ L2(P ))

such that ξ · ψρ(a) = ϕ(a) · ξ, for all a ∈ M0 and T (x) = x · ξ, for allx ∈ (Cl)∗ ⊗ ψ(M0). So we get a, possibly non-unital, ∗-homomorphismθ : M0 →M l

0 satisfying

ξψρ(a) = (id⊗ ψ)θ(a)ξ, for all a ∈M0 , (10.14)

and such that

L is the closed linear span of((Cl)∗ ⊗ ψ(M0)

)ξ . (10.15)

Put ζ = (1 ⊗ 1 ⊗ V ∗η )(1 ⊗ v∗)ξVρ. Since vv∗ = p and ξ is non-zero, we

know that ζ is non-zero. Then,

ζ ∈ (Cl ⊗ Ck ⊗Hη)H∗ρ ⊗ L2(M)

and ζ satisfies

ζ(1⊗ a) = ((id⊗ γ)θ(a))124ζ, for all a ∈M0 . (10.16)

Indeed, for all a ∈M0,

ζ(1⊗ a) = (1⊗ 1⊗ V ∗η )(1⊗ v∗)ξψρ(a)Vρ

= (1⊗ 1⊗ V ∗η )(1⊗ v∗)

(id⊗ ψ

)(θ(a)

)ξVρ, by (10.14)

= (1⊗ 1⊗ V ∗η )(id⊗ id⊗ ψη

)(id⊗ γ

)(θ(a)

)(1⊗ v∗)ξVρ, by (10.13)

=((id⊗ γ)θ(a)

)124ζ .


Define H as the closure of((Cl ⊗ Ck ⊗ Hη)∗ ⊗ M0

)ζ(Hρ ⊗ 1). Be-

cause of (10.16), H is an M0-M0-subbimodule of L2(M). By construc-tion dim(M0H) < ∞. Since N ⊂ M0 has infinite index, it follows thatM0 6≺M0 N and thus, Theorem 8.7 implies that H ⊂ L2(M0). So,

ζ ∈ (Cl ⊗ Ck ⊗Hη)H∗ρ ⊗ L2(M0) . (10.17)

In particular, ζ is G-invariant. Since ξ and v are G-invariant, we obtain,for all g ∈ G,

ζ = (id⊗ id⊗ id⊗ σg)(ζ)

= (1⊗ 1⊗ η(g)∗ ⊗ 1)(1⊗ 1⊗ V ∗η )(1⊗ v∗)ξVρ(ρ(g)⊗ 1) .

It follows that η = ρ and ζ = (ζ0)124 for some ζ0 ∈ Cl ⊗ Ck ⊗ L2(M0). Itfinally follows that

ξ = (1⊗ v)(1⊗ 1⊗ Vη)(ζ0)124V ∗η

= (1⊗ v)(id⊗ id⊗ ψη)(ζ0)

∈ (1⊗ v)(Cl ⊗ Ck ⊗⊗L2

(ψη(M0)

)), by (10.17)

= Cl ⊗K .

Because of (10.15), we conclude that L ⊂ K, ending the proof of thelemma.

Lemma 10.12. Let PHP be a finite index P -P -bimodule. For every non-zero irreducible M0-P -subbimodule H0 ⊂ H, there exists η ∈ Irr(G) anda non-zero M0-ψη(M0)-subbimodule K ⊂ H∗

η ⊗ H0 such that M0Kψη(M0)

has finite index.

Proof. For every π ∈ Irr(G), define the finite index P -P -bimodule Hπ

given by ψπ(P )(Hπ ⊗H)P . Because of Proposition 5.1, we can take ψ :P → pPnp a finite index inclusion and a P -P -bimodule isomorphism

T : PHP → ψ(P )

(p(Cn ⊗ L2(P )

))P .

Then the map 1⊗ T yields the isomorphism of P -P -bimodules

Hπ ∼= (id⊗ ψ)ψπ(P )

((1⊗ p)

(Hπ ⊗ Cn ⊗ L2(P )

))P .


Since M0 ⊂ P is irreducible, the relative commutant((id⊗ ψ)ψπ(M0)′

)∩ B(Hπ)⊗ pPnp

is finite dimensional by Lemma 4.12 and thus, Proposition 5.22 yields, forevery π ∈ Irr(G), a finite number nπ and an orthogonal decompositionHπ =

⊕nπi=1Hπ,i of Hπ into irreducible M0-P -bimodules. Lemma 10.11

yields, for every π and every i = 1, . . . , nπ,

• ηπ,i ∈ Irr(G) ,

• a ψπ(M0)-ψηπ,i(M0)-subbimodule Kπ,i ⊂ H∗ηπ,i

⊗Hπ,i

such that

• dim((Kπ,i)ψηπ,i (M0)) <∞ and ,

• if η ∈ Irr(G) and L ⊂ H∗η ⊗ Hπ,i is a non-zero ψπ(M0)-ψη(M0)-

subbimodule with dim(ψπ(M0)L) <∞ then, η = ηπ,i and L ⊂ Kπ,i.

Note that Kπ,i ⊂ HπH∗ηπ,i

⊗H.

Define the subset J ⊂ Irr(G)× Irr(G) consisting of (π, η) for which thereexists 1 ≤ i ≤ nπ with ηπ,i = η. Moreover, define

Kπ,η = spanKπ,i | ηπ,i = η .

By construction, the bimodule Kπ,η is a non-zero ψπ(M0)-ψη(M0)-subbi-module of HπH

∗η ⊗H, of finite right ψη(M0)-dimension. Let π, η ∈ Irr(G)

and K ⊂ HπH∗η⊗H, a ψπ(M0)-ψη(M0)-subbimodule of finite left ψπ(M0)-

dimension. Then,

K ⊂nπ⊕i=1

H∗η ⊗Hπ,i .

Then, every irreducible component of K is isomorphic to an irreduciblecomponent of H∗

η ⊗Hπ,i, as ψπ(M0)-ψη(M0)-bimodules. So, if we denoteby Ks the s-th irreducible component of K, we find, for every s, an index1 ≤ i(s) ≤ nπ such that Ks is isomorphic to a subbimodule ofH∗

η⊗Hπ,i(s).Then we have that η = ηπ,i(s) and Ks ⊂ Kπ,i(s). It follows that (π, η) ∈ Jand K ⊂ Kπ,η.By symmetry, we also find a subset J ′ ⊂ Irr(G)×Irr(G) and for all (π, η) ∈J ′ a ψπ(M0)-ψη(M0)-subbimodule Lπ,η of HπH

∗η⊗H which is of finite left


ψπ(M0)-dimension and which has the following property: if π, η ∈ Irr(G)and L ⊂ HπH

∗η ⊗H is a non-zero ψπ(M0)-ψη(M0)-subbimodule of finite

right ψη(M0)-dimension, we have (π, η) ∈ J ′ and L ⊂ Lπ,η.But then, J = J ′ and Kπ,η = Lπ,η for all (π, η) ∈ J = J ′. Hence, allKπ,η are finite index ψπ(M0)-ψη(M0)-bimodules. To conclude the proofof the lemma, it suffices to observe that

⊕nεi=1Hε,i is a decomposition of

H into irreducible M0-P -subbimodules and that Kε,i ⊂ H∗ηε,i

⊗Hε,i is therequired finite index M0-ψηε,i(M0)-subbimodule.

Lemma 10.13. Let PHP be a finite index P -P -bimodule and K ⊂ Ha non-zero irreducible M0-M0-subbimodule such that M0KM0 has finiteindex. Then, K is the trivial M0-M0-bimodule: M0KM0

∼= M0L2(M0)M0.

Proof. We may assume that PHP is irreducible.

Step 1. K contains a non-zero N -N -subbimodule L with dim(LN ) <∞.

By Proposition 5.1, we can take a finite index inclusion ψ : P → pPnp

such that PHP ∼= ψ(P )p(Cn ⊗ L2(P ))P . Let K ⊂ p(Cn ⊗ L2(P )) be an ir-reducible non-zero ψ(M0)-M0-subbimodule such that ψ(M0)KM0 has finiteindex. Take a finite index, irreducible inclusion ρ : M0 → qMk

0 q and aunitary

θ : q(Ck ⊗ L2(M0)) → K s.t. θ(ρ(a)µb) = ψ(a)θ(µ)b

for all µ ∈ q(Ck ⊗ L2(M0)), a, b ∈M0 .

Proposition 5.11 yields a non-zero vector ξ ∈ p(Cn(Ck)∗ ⊗ L2(P ))q suchthat ψ(a)ξ = ξρ(a) for all a ∈M0. As an element of p(Mn(C)⊗L1(P ))p,the operator ξξ∗ commutes with ψ(M0); see section 2.5. Since M0 ⊂ P

is irreducible and ψ(P ) ⊂ pPnp has finite index, the relative commu-tant ψ(M0)′ ∩ pPnp is finite dimensional, by Lemma 4.12. Hence, ξξ∗ isbounded and it follows that ξ ∈ p(Cn(Ck)∗ ⊗ P )q. The irreducibility ofρ(M0) ⊂ qMk

0 q implies that EM0(ξ∗ξ) is a non-zero multiple of q. Denote

by v ∈ p(Cn(Ck)∗ ⊗ P )q the polar part of ξ. Note that

ψ(a)v = vρ(a) for all a ∈M0 . (10.18)

We claim that ρ(N) ≺M0 N . Suppose not. Then, Theorem 8.7 impliesthat the quasi-normalizer of ρ(N) inside qMkq is contained in qMk

0 q.Since N ⊂ P is quasi-regular, it follows that

v∗ψ(P )v ⊂ qMk0 q . (10.19)


Indeed,

v∗ψ(P )v = v∗ψ(QNP (N)′′

)v

⊂ QNqMkq(v∗ψ(N)v)′′

= QNqMkq(v∗vρ(N))′′, by (10.18)

= QNqMkq(v∗vρ(N)v∗v)′′

= v∗vQNqMkq(ρ(N))′′v∗v, by Lemma 2.1

⊂ QNqMkq(ρ(N))′′, by (10.18)

⊂ qMk0 q .

Denote by A the von Neumann subalgebra of pPnp generated by ψ(P )and vv∗. Note that v∗v ∈ qMk

0 q. Since ψ(P ) ⊂ A ⊂ pPnp, it follows thatA ⊂ pPnp has finite index. Let w be an element of A. Then, we havefour different possibilities to write the word v∗wv.

1. v∗wv = v∗v(v∗ψ(a1)v)v∗v . . . v∗v(v∗ψ(an)v)v∗v ,

2. v∗wv = v∗v(v∗ψ(a1)v)v∗v . . . v∗v(v∗ψ(an)v) ,

3. v∗wv = (v∗ψ(a1)v)v∗v . . . v∗v(v∗ψ(an)v)v∗v ,

4. v∗wv = (v∗ψ(a1)v)v∗v . . . v∗v(v∗ψ(an)v) .

Because of (10.19), this proves that

v∗Av ⊂ v∗vMk0 v

∗v .

Furthermore,

v∗Av ⊂ v∗vMk0 v

∗v ⊂ v∗vP kv∗v = v∗Pnv .

We obtain finally,

[P : M0] = [v∗vP kv∗v : v∗vMk0 v

∗v]

≤ [v∗Pnv : v∗Av]

= [pPnp : A] <∞ .

We have reached a contradiction. This proves the claim.

The claim and the remark following Definition 5.21 yield b1, . . . , bm ∈q(Ck ⊗M0) such that writing V = spanbiN | i = 1, . . . ,m, we have


V 6= 0 and ρ(N)V = V . Define L ⊂ K as the closure of spanξbiN |i = 1, . . . ,m. By construction the vector ξ belongs to

((Ck)∗ ⊗ K

)q so

L is a ψ(N)-N -subbimodule of K with dim(LN ) <∞. Since EM0(ξ∗ξ) is

a multiple of q, it also follows that L is non-zero. So, Step 1 is proven.

Step 2. K is a direct sum of non-zero N -N -subbimodules L such thatNLN has finite index.

By Step 1, take a non-zeroN -N -subbimodule L0 ofK with dim(L0N ) <∞.

For all a, b ∈ QNM0(N), the closure of Na · L0 · bN is still an N -N -

subbimodule of K of finite right N -dimension. By irreducibility of M0KM0

and quasi-regularity of N ⊂M0, the linear span of all Na ·L0 ·bN is densein K. So, we have written K as a direct sum of N -N -subbimodules offinite right N -dimension. By symmetry, we can also write K as a directsum of N -N -subbimodules of finite left N -dimension. Taking all non-zero intersections of N -N -subbimodules of both kinds, we end the proofof Step 2.

Step 3. Define as above the fusion subalgebra F0 of FAlg(N) generatedby the finite index N -N -subbimodules of L2(M0). We now prove thatevery irreducible N -N -subbimodule of K belongs to F0.

Choose an infinite set Ki, i ∈ I, of irreducible, non-trivial, inequivalentN -N -bimodules that appear as N -N -subbimodules of L2(M0). Since M0

is not hyperfinite, the inclusion N ⊂ M0 has infinite index and hence,Lemma 5.16 implies that such an infinite set I can be chosen. DenoteK0i := Ki ∩M0. By Lemma 5.17, K0

i is dense in Ki and K0i is finitely

generated, both as a left and as a right N -module.

Assume by contradiction that L ⊂ K is an irreducible N -N -subbimodulesuch that NLN has finite index and L 6∈ F0. Define F as in assumption 4and note that by construction L ∈ F . Take some π ∈ Irr(G), π 6= ε. Takeη ∈ Irr(G) unitarily equivalent with the contragredient representation ofπ. Let ξ0 ∈ Hη ⊗Hπ be a non-zero (η ⊗ π)-invariant vector. Define forevery i ∈ I, the subspace Ti ⊂ P given by

Ti := span((Hη ⊗Hπ)∗ ⊗N

)(Vη)13(1⊗ 1⊗K0

i )(Vπ)23(ξ0 ⊗ 1) .

It follows from Lemma 10.9 (see the isomorphism Θ in the proof) that theclosure of Ti in L2(P ) is an N -N -bimodule isomorphic with H(ψη) ⊗NKi ⊗N H(ψπ).

Note for later use that the PTi is ultraweakly dense in P . Indeed, theultraweak closure of PTi is a left ideal, hence of the form Pq for a unique


non-zero projection q ∈ P . Since it is also a right N -module, we havethat q ∈ P ∩N ′ = C1.

We claim that the subspaces of H defined by Ti · L · P , i ∈ I, are non-zero and mutually orthogonal in H. Once this claim is proven, we havefound inside H infinitely many orthogonal, non-zero N -P -subbimodules.This is a contradiction with PHP being of finite index and N ⊂ P beingirreducible. So, to conclude step 3, it remains to prove the claim.

Fix i ∈ I. Since PTi is ultraweakly dense in P , it follows that Ti · L · Pis non-zero. By remark 10.8 and Lemma 10.9, decompose L2(P ) as adirect sum of irreducible, finite index N -N -subbimodules Hk such that,writing H0

k := Hk∩P , the linear span of all H0k is an ultraweakly dense ∗-

subalgebra of P . Take i 6= j and take k, l. We have to prove that Ti ·L·H0k

is orthogonal to Tj · L · H0l . It suffices to prove that their closures are

disjoint N -N -bimodules.

By Lemma 5.17 and our description above of the closure of Ti, the closureof Ti · L · H0

k is isomorphic with an N -N -subbimodule of

H(ψη)⊗N Ki ⊗N H(ψπ)⊗N L ⊗N Hk .

Suppose that the first letter of Hk is the non-trivial irreducible N -N -subbimodule R0 of L2(M0). Then, we have

L ⊗N Hk = L ⊗N R0 ⊗H(ψπ1)⊗N · · ·

We decompose L⊗N R0 as the direct sum of irreducible N -N -bimodulesL′i. Note that all L′i then belong to F . By Frobenius reciprocity L ⊂L′i ⊗N R0, for all i. Since we have assumed that L 6∈ F0, we have thatL′i ∈ F \ F0, for all i. So we have proven that L ⊗N Hk is isomorphicwith a direct sum of N -N -bimodules of the form

L′ ⊗N H(ψπ1)⊗N R1 ⊗N · · · ⊗N Rn−1 ⊗N H(ψπn)⊗N Rn , (10.20)

where L′ ∈ F \ F0, n ∈ N ∪ 0, π1, . . . , πn ∈ Irr(G) \ ε, R1, . . . , Rnare irreducible N -N -subbimodules of L2(M0) and R1, . . . , Rn−1 are non-trivial. Note that if Hk starts with the letter H(ψπ1), then L ⊗N Hk isalready of the form of (10.20). Hence, the closure of Ti·L·H0

k is isomorphicwith a direct sum of irreducible N -N -subbimodules that are of the form

H(ψη)⊗N Ki ⊗N H(ψπ)⊗N L′ ⊗N H(ψπ1)⊗N · · ·


with L′ ∈ F \ F0. The freeness of F and Rep(G) inside FAlg(N) impliesthat two N -N -bimodules of this last form, for different values of i, cannever be isomorphic. This proves the claim.

End of the proof. By Step 3, K contains a non-zero N -N -subbimodulethat belongs to F0. Hence, assumption 2.b says that M0KM0 is the trivialM0-M0-bimodule.

Lemma 10.14. The von Neumann algebra P is generated by

(ξ∗ ⊗ 1)ψπ(M0)(η ⊗ 1) | π ∈ Irr(G), ξ, η ∈ Hπ.

Proof. Denote by P0 the von Neumann subalgebra of P generated by(ξ∗⊗ 1)ψπ(M0)(η⊗ 1) | π ∈ Irr(G), ξ, η ∈ Hπ. Taking π = ε, note thatM0 ⊂ P0. We have to prove that P0 = P .

By Lemma 10.9, the von Neumann algebra P is densely spanned by(ξ∗ ⊗ 1)(a0)n+1(Vπ1)1,n+1 · · · (Vπn)n,n+1(an)n+1(η ⊗ 1)

∣∣∣ξ ∈

n⊗i=1

Hπi , η ∈( n⊗i=1

Hπi

)G,

where a0, . . . , an ∈M0, π1, . . . , πn ∈ Irr(G).

Therefore, it suffices to prove by induction on n that for all π1, . . . , πn ∈Irr(G), η ∈

(⊗ni=1Hπi

)Gand a0, . . . , an ∈M0, we have

An := (a0)n+1(Vπ1)1,n+1 · · · (Vπn)n,n+1(an)n+1(η⊗1) ∈ Hπ1⊗· · ·⊗Hπn⊗P0.

(10.21)

The case n = 1 being trivial, assume that (10.21) holds for all n ≤ k− 1.Take Ak as in (10.21) and re-write Ak in the following way.

Ak = (a0)k+1(ψπ1(a1))1,k+1(Vπ1)1,k+1(Vπ2)2,k+1 · · · (ak)k+1(η ⊗ 1) .

Lemma 7.15 yields µ1, . . . , µr ∈ Irr(G), isometries vi ∈ Mor(µi, π1 ⊗ π2)and Xi ∈ B(Hµi ,Hπ1 ⊗Hπ2)⊗N such that

(Vπ1)13(Vπ2)23 =r∑i=1

XiVµi(v∗i ⊗ 1) .


Put ξi := (v∗i )12 η ∈ (Hµi ⊗Hπ3 ⊗ · · · ⊗Hπk)G and

Bi := (Vµi)1,k(a2)k(Vπ3)2,k(Vπk)k−1,k(ak)k(ξi ⊗ 1) .

By the induction hypothesis, Bi ∈ Hµi ⊗⊗k

i=3Hπi ⊗ P0, for all i. Also,a0 ∈ P0 and ψπ1(a1) ∈ B(Hπ1)⊗P0. Since Xi ∈ B(Hµi ,Hπ1 ⊗Hπ2)⊗P0,it follows that

Ak = (a0)k+1(ψπ1(a1))1,k+1

r∑i=1

(Xi)1,2,k+1Bi ∈ Hπ1 ⊗ · · · ⊗Hπk⊗ P0 .

So, the lemma is proven.

We can finally prove Theorem 10.7.

Proof. Since the functor

Rep(G) → Bimod(P ) : π 7→ PMor(L2(M),Hπ)P

is a fully faithful tensor functor, and since

PMor(L2(M),Hπ)P ∼= PMor(Hπ,L2(M))P ∼= ψπ(P )(Hπ ⊗ L2(P ))P

as P -P -bimodules, it suffices to prove that for every irreducible finiteindex P -P -bimodule PHP , there exists an η ∈ Irr(G) such that PHP ∼=ψη(P )(Hη ⊗ L2(P ))P . So, let PHP be an irreducible finite index P -P -bimodule.

Decompose H into a direct sum H =⊕k

i=1Hi of irreducible M0-P -subbimodules. By Lemma 10.12, we can take η1, . . . , ηk ∈ Irr(G) andnon-zero irreducible M0-ψηi(M0)-subbimodules Ki ⊂ H∗

ηi⊗Hi such that

M0Kiψηi (M0) has finite index. Viewing Ki as an M0-ψηi(M0)-subbimoduleof the finite index bimodule P(H∗

ηi⊗H)ψηi (P ), Lemma 10.13 says that

M0Kiψηi (M0)∼= M0L

2(M0)M0.

By Proposition 5.1 we can take a finite index inclusion ψ : P → pPnp

and a P -P -bimodule isomorphism

PHP ∼= ψ(P )

(p(Cn ⊗ L2(P )

))P .

Denote A = ψ(M0)′ ∩ pPnp. Then, A is finite dimensional by Lemma4.12 and Hi corresponds to pi(Cn⊗L2(P )), where p1, . . . , pk are minimalprojections in A summing to p.


In the previous paragraph, it was shown that

ψ(M0)pi

(pi(CnH∗

ηi⊗ L2(P )

))ψηi (M0)

contains the trivial M0-ψηi(M0)-bimodule. So, we can take non-zero vec-tors vi ∈ pi(CnH∗

ηi⊗ L2(P )) satisfying ψ(a)vi = viψηi(a) for all a ∈ M0.

As an element of pi(Mn(C)⊗L1(P ))pi, the operator viv∗i commutes withψ(M0); see section 2.5. By minimality of pi, it follows that viv∗i is a mul-tiple of pi and in particular, vi ∈ pi(CnH∗

ηi⊗P ). So, we may assume that

viv∗i = pi. On the other hand, since M ′

0 ∩M = C1, a similar proof as inLemma 7.18 implies that B(Hηi)⊗ P ∩ ψηi(M0)′ = C1. So, v∗i vi = 1.

Define η =⊕k

i=1 ηi and put

ψη : P → B(Hη)⊗ P : ψη(a) =k⊕i=1

ψηi(a) .

Note that ψη(a) = Vη(1 ⊗ a)V ∗η , where Vη =

⊕ki=1 Vηi . We have shown

the existence of u ∈ p(CnH∗η ⊗ P ) satisfying uu∗ = p, u∗u = 1 and

u∗ψ(a)u = ψη(a) for all a ∈M0. We may assume from now on that

PHP ∼= ψ(P )(Hη ⊗ L2(P ))P , (10.22)

where ψ : P → B(Hη) ⊗ P is a finite index inclusion satisfying ψ(a) =ψη(a) for all a ∈M0. It remains to prove that ψ(a) = ψη(a) for all a ∈ P .

By Lemma 10.14, it suffices to prove that (id⊗ψ)ψπ(a) = (id⊗ψη)ψπ(a)for all a ∈M0 and all π ∈ Irr(G). Fix π ∈ Irr(G).

Let T : PHP → ψπ(P )

(Hη ⊗ L2(P )

)P be the isomorphism of P -P -bimodu-

les obtained in (10.22). Then, 1⊗ T yields an isomorphism

ψπ(P )

(Hπ ⊗H

)P∼= (id⊗ ψ)ψπ(P )

(Hπ ⊗Hη ⊗ L2(P )

)P .

Applying the reasoning in the previous paragraphs to the P -P -bimoduleψπ(P )(Hπ ⊗H)P , we find a finite dimensional unitary representation γ ofG and a unitary

X ∈ B(Hγ ,Hπ ⊗Hη)⊗ P ,

satisfying(id⊗ ψ)ψπ(a) = Xψγ(a)X∗ (10.23)

for all a ∈M0.


Because ψπ(N) ⊂ B(Hπ)⊗N and ψ(a) = ψη(a) for all a ∈ N , it followsin particular that (id⊗ ψη)ψπ(a) = Xψγ(a)X∗ for all a ∈ N . Hence, theunitary

Z := (Vπ)∗13(Vη)∗23XVγ

satisfies Z(1 ⊗ a) = (1 ⊗ 1 ⊗ a)Z for all a ∈ N . So, Z = U ⊗ 1, whereU : Hγ → Hπ ⊗Hη. Note that Z is a G-invariant unitary and hence, Uintertwines the representations γ and π⊗η. It follows that for all a ∈M0,

(id⊗ ψη)ψπ(a) = (Vη)23(Vπ)13(1⊗ 1⊗ a)(Vπ)∗13(Vη)∗23

= XVγZ∗(1⊗ 1⊗ a)ZV ∗

γ X∗

= XVγ(U∗U ⊗ a)V ∗γ X

∗

= Xψγ(a)X∗ .

So we have that Xψγ(a)X∗ = (id⊗ ψη)ψπ(a) for all a ∈M0. Combiningwith (10.23), we conclude that (id ⊗ ψ)ψπ(a) = (id ⊗ ψη)ψπ(a) for alla ∈M0.

As a consequence of Theorem 10.7, we now prove Theorem 10.1 stated inthe introduction.

Proof of Theorem 10.1. Denote by M1 the hyperfinite II1 factor and takea minimal action G y M1. Put N := MG

1 .

Define the group Γ, its subgroup Λ < Γ and the scalar 2-cocycle Ω ∈Z2(Γ, S1) as in Example 10.6. Write R := LΩ(Λ) and M0 := LΩ(Γ).Denote by F the fusion subalgebra of FAlg(R) generated by all finiteindex R-R-bimodules appearing as an R-R-subbimodule of a finite indexM0-M0-bimodule. By Remark 10.4, F is countable.

Note that both N and R are isomorphic with the hyperfinite II1 factor.Whenever α : N → R is an isomorphism, we can view α−1Fα as a fusionsubalgebra of FAlg(N). By Theorem 6.15, we can choose α such thatα−1Fα is free w.r.t. the image of Rep(G) inside FAlg(N). Identifying Nand R through α, it follows from Example 10.6 that all assumptions forTheorem 10.7 are satisfied.

So, we can take M := M0 ∗N M1, extend G y M1 to a minimal actionG y M by acting trivially on M0 and conclude from Theorem 10.7 thatthe natural tensor functor Rep(G) → Bimod(MG) is an equivalence ofcategories.


10.3 Proof of Theorem 10.2

Fix a second countable compact group G and a minimal action Gσy M on

a II1 factor M . Let α : G y A be an action of G on the finite dimensionalvon Neumann algebra A. Denote by α ⊗ σ the diagonal action of G onA ⊗M , given by (α ⊗ σ)g = αg ⊗ σg for all g ∈ G. Denote by Nβ thefixed point algebra of an action β on a von Neumann algebra N .

Lemma 10.15. Using the foregoing notations, we have that

(A⊗M)α⊗σ is a factor ⇔ Z(A)α = C1 .

Furthermore,

1⊗Mσ ⊂ (A⊗M)α⊗σ is irreducible ⇔ Aα = C1 .

Every intermediate von Neumann algebra 1⊗Mσ ⊂ N ⊂ (A⊗M)α⊗σ isof the form (D ⊗M)α⊗σ for a uniquely determined globally G-invariant∗-subalgebra D ⊂ A.

Proof. Denote P := Mσ. By minimality, the relative commutant of 1⊗Pinside (A⊗M)α⊗σ equals Aα⊗ 1. So, the inclusion 1⊗P ⊂ (A⊗M)α⊗σ

is irreducible iff Aα = C1. Also, Z((A⊗M)α⊗σ

)⊂ Aα ⊗ 1.

We claim that

A = span(id⊗ ω)(a) | a ∈ (A⊗M)α⊗σ , ω ∈M∗ . (10.24)

Let π ∈ Irr(G). We saw in the proof of Proposition 7.6 that we can takenon-zero elements X ∈ H∗

π ⊗A such that (id⊗αg)(X) = X(π(g)⊗ 1) forall g ∈ G. Since A is finite dimensional, the linear span of all possibleX(Hπ ⊗ 1) is a finite dimensional vector space. Proposition 7.6 impliesthat A is exactly the linear span of all X(Hπ ⊗ 1).

On the other hand an immediate computation shows that X12(Vπ)∗13 be-longs to H∗

π ⊗ (A⊗M)α⊗σ so X12 ⊂ H∗π ⊗

((A⊗M)α⊗σ(1⊗M)

). Then,(

X(Hπ ⊗ 1))⊗ 1 = X12(Hπ ⊗ 1⊗ 1) ⊂ (A⊗M)α⊗σ(1⊗M) ,

which implies that

X(Hπ ⊗ 1) = (id⊗ τ)((X(Hπ ⊗ 1)

)⊗ 1)

⊂ (id⊗ τ)((A⊗M)α⊗σ(1⊗M)

)⊂ span(id⊗ ωx)(a) | a ∈ (A⊗M)α⊗σ, x ∈M ,


where ωx(y) = τ(yx), for all y ∈ M . This proves that X(Hπ ⊗ 1) isincluded in the expression at the right-hand side of (10.24). So (10.24) isproven.

A combination of the claim and the first paragraph of the proof impliesthat Z

((A ⊗M)α⊗σ

)= Z(A)α ⊗ 1. One inclusion is obvious, because

Z(A)α⊗1 ⊂ (A⊗M)α⊗σ ∩(Z(A)⊗1

). From the previous paragraph we

know that Z((A⊗M)α⊗σ

)⊂ Aα ⊗ 1. So, if x ∈ Z

((A⊗M)α⊗σ

), write

x = y ⊗ 1, with y ∈ Aα. Then

(y ⊗ 1)a = a(y ⊗ 1), for all a ∈ (A⊗M)α⊗σ .

Applying (id⊗ ω) to the previous equality, for ω ∈M∗ yields

y(id⊗ ω)(a) = (id⊗ ω)(a)y, for all a ∈ (A⊗M)α⊗σ ,

and by (10.24) it follows that y ∈ Z(A). Hence, (A ⊗M)α⊗σ is a factoriff Z(A)α = C1.

Let 1⊗Mσ ⊂ N ⊂ (A⊗M)α⊗σ be an intermediate von Neumann algebra.Choose a G-invariant trace on A and we view A as a finite-dimensionalHilbert space, with inner product given by this invariant trace.

In this way, α acts on A in a trace-preserving way and can thus be seenas a finite dimensional unitary representation πA : G → U(A). Therepresentation πA is not irreducible but as was done in the beginningof the proof of Theorem 7.24 (see formula (7.12)), we can still choose aunitary W ∈ B(A) ⊗M satisfying (id ⊗ σg)(W ) = W (πA(g) ⊗ 1) for allg ∈ G. Define the finite index inclusion γ : P → B(A) ⊗ P : γ(a) =W (1⊗ a)W ∗. The map a 7→Wa defines a P -P -bimodular unitary

Θ : (1⊗ P )L2((A⊗M)α⊗σ

)(1⊗ P ) → γ(P )

(A⊗ L2(P )

)P .

Viewing N as a P -P -subbimodule of (1⊗ P )L2((A⊗M)α⊗σ

)(1⊗ P ), its im-

age under Θ(N) becomes a P -P -subbimodule of γ(P )

(A⊗ L2(P )

)P . Thus,

Θ(N) = q(A ⊗ L2(P )), where q is a projection in B(A) ⊗ P ∩ γ(P )′. Infact Θ(N) = q(A⊗ P ). Moreover, we have that

B(A)⊗ P ∩ γ(P )′ = W (B(A)AdπA ⊗ 1)W ∗ . (10.25)

Indeed, let T ∈ B(A) ⊗ P ∩ γ(P )′. The element W ∗TW belongs toB(A) ⊗ 1, by minimality, and becomes then G-invariant. Write T =


W (X ⊗ 1)W ∗, with X ∈ B(A). The G-invariance of T implies that X is(AdπA)-invariant. So, (10.25) is proven.

Write q = W (p⊗1)W ∗ and define the vector subspaceD ⊂ A as the imageof the projection p. Since p commutes with πA(G), it follows that D isglobally α-invariant. Then, p ⊗ 1 = Θ∗qΘ is a projection in B(A) ⊗M ,restricted to the subspace (A⊗M)α⊗σ. So we have

(p⊗ 1)((A⊗M)α⊗σ

)= Θ∗qΘ

((A⊗M)α⊗σ

)= Θ∗q(A⊗ P ) = N

Moreover,

(p⊗ 1)((A⊗M)α⊗σ

)= Im

((p⊗ 1)|(A⊗M)α⊗σ

)= (A⊗M)α⊗σ ∩ (D ⊗M)

= (D ⊗M)α⊗σ .

We have shown that N = (D ⊗M)α⊗σ. It remains to prove that D is a∗-algebra.

SinceD is globally α-invariant, we can seeG y D, via the restriction of α.Then, we may view α as a unitary representation α : G→ B(D) ⊂ B(A).We decompose the representation α into irreducible representations asα =

⊕π∈Irr(G) π. So we have a family of isometries Xπ : Hπ → D such

that∑

π∈Irr(G)XπX∗π = 1 and satisfying

Xπ(π(g)ξ) = αg(Xπξ), for all ξ ∈ Hπ .

It follows that

D = spanXπ(ξ) | ξ ∈ Hπ, π ∈ Irr(G) .

If we view the isometries Xπ : Hπ → D as elements of H∗π ⊗D we have

that D is linearly spanned by elements of the form X(Hπ ⊗ 1), whereπ ∈ Irr(G), X ∈ H∗

π ⊗D and (id ⊗ αg)(X) = X(π(g) ⊗ 1) for all g ∈ G.As in (10.24), it follows that D is linearly spanned by elements of theform (id⊗ ω)(a), with ω ∈M∗ and a ∈ N . Hence, D = D∗.

Further, let π, η ∈ Irr(G), X ∈ H∗π ⊗D, Y ∈ H∗

η ⊗D and (id⊗αg)(X) =X(π(g) ⊗ 1), (id ⊗ αg)(Y ) = Y (η(g) ⊗ 1) for all g ∈ G. To conclude theproof of the lemma, it suffices to show that X13Y23 ∈ (Hπ ⊗ Hη)∗ ⊗ D

because then,

X(Hπ ⊗ 1)Y (Hη ⊗ 1) = X13Y23(Hπ ⊗Hη ⊗ 1) ⊂ D .


But, we know that X12(Vπ)∗13 ∈ H∗π⊗N and Y12(Vη)∗13 ∈ H∗

η⊗N . Viewingthese two elements in B(Hπ) ⊗ B(Hη) ⊗D ⊗M and using the fact thatN is an algebra, we have that

X13 (Vπ)∗14 Y23 (Vη)∗24 ∈ (Hπ ⊗Hη)∗ ⊗N ⊂ (Hπ ⊗Hη)∗ ⊗D ⊗M .

The two factors in the middle commute and the conclusion follows.

We are now ready to prove Theorem 10.2.

Proof of Theorem 10.2. Take G y M as in the formulation of the the-orem and put P := Mσ. Let P0 ⊂ P be a finite index subfactor. Wefirst prove that P0 is unitarily conjugate in P to a subfactor of the formP (α) for some action G

αy A of G on a finite dimensional von Neumannalgebra A satisfying Z(A)α = C1.

Let P0 ⊂ P ⊂ P1 be the basic construction. Then, PL2(P1)P is a finiteindex P -P -bimodule. By assumption, we find a finite dimensional unitaryrepresentation π : G → U(n) and a unitary V ∈ Mn(C) ⊗M satisfying(id⊗ σg)(V ) = V (π(g)⊗ 1) for all g ∈ G, such that

PL2(P1)P ∼= γ(P )

(Cn ⊗ L2(P )

)P , where γ(a) = V (1⊗a)V ∗ for all a ∈ P .

The left P1-action on L2(P1) commutes with the right P -action and hence,we can extend γ to an inclusion γ : P1 → Mn(C) ⊗ P . Denote N =V ∗γ(P1)V and write α(g) = Ad(π(g)). Then α defines an action byautomorphisms of G on Mn(C) and we have 1 ⊗ P ⊂ N ⊂ (Mn(C) ⊗M)α⊗σ. The first inclusion is obvious since P ⊂ P1. Take an elementx ∈ P1, we have, for all g ∈ G,

(αg ⊗ σg)(V ∗γ(x)V )

= (αg ⊗ id)(id⊗ σg)(V ∗γ(x)V )

=(Ad(π(g)⊗ 1)

)((π(g)∗ ⊗ 1)V ∗γ(x)V (π(g)⊗ 1)

)= V ∗γ(x)V ,

which proves the second inclusion. Applying Lemma 10.15, we find afinite dimensional α-invariant von Neumann algebra A ⊂ Mn(C) suchthat N = (A⊗M)α⊗σ. The group G acts on A via α, by restriction, andZ(A)α = C1. Moreover, viewing γ as an isomorphism γ : P1 → γ(P1),we obtain the ∗-isomorphism

θA := (AdV ) γ : P1 → (A⊗M)α⊗σ ,


satisfying θA(a) = 1 ⊗ a for all a ∈ P . So θA yields an isomorphism ofinclusions (

1⊗ P ⊂ (A⊗M)α⊗σ) ∼= (P ⊂ P1

).

By uniqueness of the tunnel construction, it follows that P0 and P (α) areunitarily conjugated in P .

Finally, suppose that Gαy A and G

βy B satisfy Z(A)α = C1 and

Z(B)β = 1 and suppose that the subfactors P (α) and P (β) are unitarilyconjugate in P . It remains to construct a ∗-isomorphism γ : A → B

satisfying βg γ = γ αg for all g ∈ G. By assumption, we find a∗-isomorphism

θ := θB θ−1A : (A⊗M)α⊗σ → (B ⊗M)β⊗σ ,

satisfying θ(1⊗a) = 1⊗a for all a ∈ P . Repeating the argument given inthe proof of Lemma 10.15, we have the following data, for all C ∈ A,B.

• A finite dimensional representation πC : G→ U(C) .

• A unitary WC ∈ B(C)⊗M satisfying (id⊗σg)(WC) = WC(πC(g)⊗1).

• A ∗-homomorphism γC : P → B(C)⊗P satisfying γC(x) = WC(1⊗x)W ∗

C .

• A P -P -bimodular unitary ψC : L2((A ⊗ M)α⊗σ

)→ A ⊗ L2(P )

satisfying ψC(a) = WCa .

Then we have the following situation.

A⊗ L2(M) θ // B ⊗ L2(M)

1⊗ PL2((A⊗M)α⊗σ

)1⊗ P

?

OO

θ //

ψA

1⊗ PL2((B ⊗M)β⊗σ

)1⊗ P

?

OO

ψB

γA(P )

(A⊗ L2(P )

)P

T //γB(P )

(B ⊗ L2(P )

)P

The map T is a unitary intertwiner so by Corollary 5.13, there exists aunitary Y ∈ B(A,B)⊗P such that T (ξ) = Y ξ, for all ξ ∈ A⊗L2(P ) andsatisfying

Y γA(x) = γB(x)Y, for all x ∈ P . (10.26)


The map θ is given by the operator W ∗BYWA which belongs to B(A,B)⊗

M and commutes with 1 ⊗ P , by (10.26). So we get a bijective mapγ : A → B such that θ and γ ⊗ id coincide on A ⊗M . In particular,θ = (γ ⊗ id)|(A⊗M)α⊗σ and

W ∗BYWAa = (γ ⊗ id)(a), for all a ∈ (A⊗M)α⊗σ . (10.27)

The elementW ∗BYWA is σ(G)-invariant so we have that (πB(g)⊗1)

(W ∗BY

WA

)=(W ∗BYWA

)(πA(g)⊗ 1), for all g ∈ G. Because of (10.27) and the

definition of πA and πB, we obtain that βg γ = γ αg, for all g ∈ G.

We prove that γ is a ∗-homomorphism. We apply the same reasoningas in the proof of Lemma 10.15. Since G acts on the finite dimensionalalgebra A via α we can take, for every π ∈ Irr(G), non-zero elementsX ∈ H∗

π ⊗ A satisfying (id ⊗ αg)(X) = X(π(g) ⊗ 1) for all g ∈ G, suchthat A is exactly the linear span of all X(Hπ ⊗ 1). We also have that

X12(Vπ)∗13 ∈ H∗π ⊗ (A⊗M)α⊗σ .

Let π, η ∈ Irr(G), X ∈ H∗π ⊗ A, Y ∈ H∗

η ⊗ A and (id ⊗ αg)(X) =X(π(g) ⊗ 1), (id ⊗ αg)(Y ) = Y (η(g) ⊗ 1) for all g ∈ G. Since θ is ahomomorphism, we obtain that

(id⊗ id⊗ θ)((X13(V ∗

π )14Y23(V ∗η )24

)=(

(id⊗ θ)(X12(Vπ)∗13

))134

((id⊗ θ)(Y12(V ∗

η )13))

234. (10.28)

Since the middle terms (V ∗π )14 and Y23 commute and since θ(a) = (γ ⊗

id)(a), for all a ∈ (A⊗M)α⊗σ, the left-hand side of (10.28) is equal to((id⊗ id⊗ γ)(X13Y23

)⊗ 1)(V ∗π )14(V ∗

η )24 . (10.29)

The right-hand side of (10.28) is equal to((id⊗ γ)(X)

)13

(V ∗π )14

((id⊗ γ)(Y )

)23

(V ∗η )24 . (10.30)

We deduce, from (10.29) and (10.30) that

(id⊗ id⊗ γ)(X13Y23) =((id⊗ γ)(X)

)13

((id⊗ γ)(Y )

)23,

the multiplicativity of γ follows.


We prove now that γ is a ∗-homomorphism. We have proven in (10.24)that A is linearly spanned by elements of the form (id⊗ω)(a), for ω ∈M∗and a ∈ (A⊗M)α⊗σ. We easily see that γ satisfies

γ((id⊗ ω)(a)

)= (id⊗ ω)(θ(a)) .

Since θ is a ∗-homomorphism, it follows that γ(a∗) = γ(a)∗, for all a ∈A.

Appendix A

Functor Rep(G) → Bimod(P )

Let G y M be a minimal action and P := MG. We consider the functor

F : Rep(G) → Bimod(P ) : π 7→ PMor(L2(M),Hπ)P .

The aim of this appendix is to prove that the map

Mor(π, η) → Mor(F (π), F (η)) : Ω 7→ F (Ω)

is an isomorphism, for every π, η ∈ Rep(G).

Recall that we have maps ψπ : P → B(Hπ) ⊗ P such that ψπ(x) =Vπ(1 ⊗ x)V ∗

π , defined for every finite dimensional representations π. SeeChapter 7 for the notations. We use the notation H(ψπ) to denote thebimodule P

((H∗

π ⊗ L2(P ))ψπ(P ). We denote by Rv the operator of right

multiplication by v.

In the proof of Theorem 7.24, we have proven that the maps

Mor(π, η)


Cm(Cn)∗ ⊗ P

v 7→Rv∗


yield an isomorphism Mor(π, η) ∼= MorP (H(ψπ),H(ψη))P .

222 Chapter A. Functor Rep(G) → Bimod(P )

In Lemma 7.20, we proved that the map

∆π : H∗π ⊗ P → Mor(Hπ,L2(M)) ,

such that ∆π(x)ξ := xVπ(ξ ⊗ 1), for all ξ ∈ Hπ, is an isomorphism. Wecan give an explicit formula for the inverse of ∆π. We use the followingnotations, already introduced in Proposition 7.6.

Notation A.1. We represent every S ∈ Mor(Hπ,L2(M)) by an elementQS ∈ H∗

π ⊗ L2(M) defined by QS(ξ ⊗ 1) = S(ξ), for every ξ ∈ Hπ.

Then, it is easy to check that

(∆π)−1(S) = QSV∗π , for all S ∈ Mor(Hπ,L2(M)) .

We have the following isomorphisms.

Mor(π, η)


Cm(Cn)∗ ⊗ P

v 7→Rv∗


T 7→∆ηT(∆π)−1

MorP (PMor(Hπ,L2(M))P , PMorP (Hη,L2(M))P)P

Θ (see definition below)

MorP (F (π), F (η))P

The isomorphism Θ is defined by

Θ(ϕ)(S) = (ϕ S∗)∗ ,

for all ϕ ∈ MorP (PMor(Hπ,L2(M))P , PMorP (Hη,L2(M))P)P and S ∈PMor(L2(M),Hπ)P .

Let Ω ∈ Mor(π, η). The element

T (Ω) = ∆η RVπ(Ω∗⊗1)V ∗η (∆π)−1

)

223

belongs to MorP (PMor(Hπ,L2(M))P , PMorP (Hη,L2(M))P)P . We provethat F (Ω) = Θ

(T (Ω)

)which implies that

Mor(π, η) ∼= MorP (F (π), F (η))P .

We first compute, for every S ∈ Mor(Hπ,L2(M)),

T (Ω)(S) =(∆η RVπ(Ω∗⊗1)V ∗

η (∆π)−1

)(S)

=(∆η RVπ(Ω∗⊗1)V ∗

η

)(QSV ∗

π )

= ∆η

(QS(Ω∗ ⊗ 1)V ∗

η

).

Then, for every ξ ∈ Hη,

T (Ω)(S)ξ = QS(Ω∗ξ ⊗ 1)

= (S Ω∗)ξ .

So we obtain, that for every S ∈ Mor(L2(M),Hπ),

Θ(T (Ω)

)(S) =

(T (Ω)(S∗)

)∗= (S∗ Ω∗)∗

= Ω S= F (Ω)(S) .

224 Chapter A. Functor Rep(G) → Bimod(P )

Appendix B

Frobenius reciprocity

theorem

LetH be anM -N -bimodule, K anN -P -bimodule and L anM -P -bimodu-le. We also assume that H, K and L all have finite index. In Theorem5.28, we defined the isomorphism

Θ : MorM (H⊗N K,L)P → MorN (K,H⊗M L)P : T 7→ (1⊗ T )(RH ⊗ 1) .

We prove here that Θ is natural in H,K and L, which will follow fromthe next lemma. We use the notations of section 5.6.

Lemma B.1. Denoting λξ : K → H ⊗N K and λξ : L → H ⊗M L, weobtain that

λ∗ξΘ(T ) = T λξ, for all ξ ∈ H0 .

Proof. For every η ∈ K and µ ∈ L,

〈(λ∗ξΘ(T )

)η, µ〉 = 〈(RH ⊗ 1)η, ξ ⊗ T ∗(µ)〉

= 〈ξ ⊗ η, T ∗(µ)〉, by Lemma 5.27

= 〈(T λξ)η, µ〉 .

226 Chapter B. Frobenius reciprocity theorem

We prove that Θ is natural in L. Let S ∈ MorM (L,L′)P . Then, thefollowing diagram commutes.

MorM (H⊗N K,L)PT 7→Θ(T )//

T 7→ST

MorN (K,H⊗M L)P

T 7→(1H⊗S)T

MorM (H⊗N K,L′)PT 7→Θ(T )// MorN (K,HM ⊗ L′)P

We have to prove that Θ(S T ) = (1H⊗S) Θ(T ). It is enough to provethat this equality holds, up to composing on the left by λ∗

ξ. Because of

Lemma B.1, we only need to prove that S λ∗ξ

= λ∗ξ (1H ⊗ S). We

compute

〈(S λ∗ξ)η, µ〉 = 〈η, ξ ⊗ S∗(µ)〉 = 〈(λ∗

ξ (1H ⊗ S))η, µ〉 .

We prove now that for every T ∈ MorM (H ⊗N K,L)P , we have Θ(T (S ⊗ 1K)) = (S∗ ⊗ 1L) Θ(T ), for every S ∈ MorN (H′,H)N and thus, Θis natural in H. Here again, we only need to prove this equality up tocomposing on the left by λξ, for every ξ ∈ H0. Then, for every η ∈ K

and µ ∈ L,

〈(λ∗ξ (S∗ ⊗ 1L) Θ(T )

)η, µ〉 = 〈Θ(T )η, S(ξ)⊗ µ〉

= 〈Θ(T )η, S(ξ)⊗ µ〉= 〈

(λ∗S(ξ)

Θ(T ))η, µ〉

= 〈(T λS(ξ)

)η, µ〉

= 〈(T (S ⊗ 1K) λξ

)η, µ〉

=⟨(λ∗ξΘ(T (S ⊗ 1K)

))η, µ⟩.

We conclude the proof, by proving the naturality in K. We prove thatΘ(T (1H ⊗ S)) = Θ(T ) S, for every S ∈ MorN (K ′

,K)P . For everyξ ∈ H0, we have

λ∗ξΘ(T (1H ⊗ S)

)= T (1H ⊗ S) λξ= T λξ S= λ∗

ξΘ(T ) S .

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Index of keywords

C∗-tensor category, 115C∗-algebra, 18Banach algebra, 17adjoint, 17atom, 23bounded operators, 17central support, 22commutant, 19correspondences, 38diffuse, 23factors, 21final projection, 22initial projection, 22involution, 18isometry, 18left support projection, 22minimal projection, 23normality, 20partial isometry, 22polar decomposition, 22projections, 21right support projection, 22von Neumann algebras, 19

affiliated operator, 44almost normalizing bimodules, 110amalgamated free product, 151amenable, 27, 39anti-representation, 38

basic construction, 68bicommutant, 19bimodule, 38, 49bimodule dimension, 93bounded vectors, 51, 76

coboundary, 34cocycle, 34conditional expectation, 27conjugate actions, 31conjugates, 89Connes’ tensor product, 51, 74contragredient, 58coupling constant, 64crossed product, 28

equivalent, 22ergodic, 29, 30essentially free, 30

faithful, 24finite projection, 23free, 29freeness, 113Frobenius reciprocity, 89fusion algebra, 101

GNS construction, 25group von Neumann algebra, 26

hyperfinite, 27

INDEX OF KEYWORDS 233

ICC group, 26injective von Neumann algebra,

27intertwiner, 49, 59, 79intertwining bimodules, 84

Jones index, 66

left M -module, 38

measure preserving action, 30minimal action, 130module dimension, 63

orbit equivalent actions, 31outer action, 41

Pimsner-Popa basis, 74predual, 19Property (T), 38

relative property (T), 38right N -module, 38

spectral subspace, 122state, 20strong topology, 18sub-equivalent, 22

twisted group von Neumann al-gebras, 34

ultrastrong topology, 18ultraweak topology, 18

von Neumann equivalent actions,31

weak topology, 18

234 INDEX OF KEYWORDS

Index of symbols

[A] Closed vector space generated by A, 11αω Automorphism of N o Γ such that α(ug) = ω(g)ug,

44A ≺M B A is embedded into B inside M , 84A1 ∗ A2 Free product of the fusion algebras A1 and A2, 114Aut(N ⊂M) Automorphisms of M leaving N globally invariant,

43

B(H) Algebra of bounded operator on H, 11B(H,K) Vector space of bounded operators from H to K, 11Bimod(M) Category of finite index bimodules over the II1 factor

M , 115

Cn(Cm)∗ Bounded operators B(Cm,Cn), 12Char(G) One-dimensional unitary representations of the com-

pact group G, 36Char(Γ) Group of characters of the abelian countable group

Γ, 36CommΓ(Λ) Commensurator of the subgroup Λ in Γ, 34Cπξ,η Coefficient (ξ, η) of the representation π, 120Cπjk Coordinate functions, 120

δi,j Kronecker symbol, 11∂2(Γ, S1) S1-valued coboundaries on Γ, 35∂a Boundary of the map a : G→ S1, 35dim(HM) Dimension of the right module HM, 64dim(MH) Dimension of the left M -module MH, 65dimα Dimension function on finite index bimodules, 98

236 INDEX OF SYMBOLS

eij Matrix units, 12EM Conditional expectation onto M , 68EM Conditional expectation onto M , 27eN Jones’ projection, 68ε The trivial representation, 119

F0 Fusion subalgebra of FAlg(N) generated by all finiteindexN -N -subbimodules of L2(M), 108

Fα Fusion algebra consisting of bimodules Kα, 114FAlg(M) The fusion algebra of the II1 factor M , 103Fn Free group on n generators, 26

grp(A) Intrinsic group of the fusion algebra A, 102

H0(s) Nus, 175H0 Right bounded vectors of H, 510H Left bounded vectors of H, 53H2(Γ, S1) 2-cohomology group of Γ, 35H Contragredient Hilbert space, 11Hπ Hilbert space of the representation π, 119H(ψ) Generic form of a finite index bimodule, 73Hrep(π) Bimodule M

((Ck)∗ ⊗ L2(M)

)θπ(M) with θπ(aug) =

π(g)⊗ aug, 107H Orthogonal complement, 152H(s) The closure of Nus, 175

N〈·, ·〉 Left N -valued inner product, 54〈·, ·〉N Right N -valued inner product, 51Irr(G) Set of two by two inequivalent and irreducible unitary

representations of G, 133

JM Canonical anti-unitary on L2(M, τ), 26

Kα Conjugation of the bimodule K by the automorphismα, 114

INDEX OF SYMBOLS 237

L0(π) Linear span of Mor(Hπ,M)Hπ, 123L2(M, τ) GNS construction, 26LΩ(Γ) Group von Neumann algebra of Γ twisted by the 2-

cocycle Ω, 36Lξ Densely defined operator of left multiplication by ξ,

51λΩ Left Ω-regular representation, 36λτ Operator of left multiplication on L2(M, τ), 26L(Γ) Group von Neumann algebra of the countable group

Γ, 26L(π) Closure of L0(π) inside L2(M), 123

M ′ Commutant of M , 19M ′′ Bicommutant of M , 19M0 ∗N M1 Amalgamated free product of M0 and M1 over N ,

154Mn Mn(C)⊗M , 12Mop Opposite algebra of M , 38M∗ Predual of M , 19〈M, eN 〉 Basic construction, 68MG Fixed-points under the action of the group G on M ,

123[M : N ] Jones’ index of the inclusion N ⊂M , 66Mor(π, ρ) Intertwiner T : Hπ → Hρ, 119Mor(Hπ,M) Linear maps S : Hπ →M satisfying σg S = S π(g),

123Mor(Hπ,L2(M)) T : Hπ → L2(M) such that σg(T (ξ)) = T (π(g)ξ), 126MorM (H,K)N M -N -bimodular operators from H to K, 49MorM (H)N M -N -bimodular operators on H, 49M o Γ crossed product of M by the countable group Γ, 29Mσ Fixed-points under the action σ on M , 123

N[G] Fusion algebra with basis G, 101NormM (N) Normalizer of N inside M , 33

Ωπ Obstruction cocycle of π, 35∧i∈I pi Infimum projection, 21∨i∈I pi Supremum projection, 21

238 INDEX OF SYMBOLS

p ∼ q The projection p is equivalent to q, 22p . q The projection p is sub-equivalent to q, 22π ∼ ρ equivalent representations, 120π Contragredient representation, 120ψπ ψπ(a) = Vπ(1⊗ a)V ∗

π , 135

QNM (N) Quasi-normalizer of N inside M , 33

Rξ Densely defined operator of right multiplication by ξ,53

(RH, RH) Pair of conjugates, 90, 116Rep(G) Category of finite dimensional representations of

the compact group G, 116ρτ Operator of right multiplication on L2(M, τ), 26

S1 Complex numbers of modulus 1, 34S∞ Infinite symmetric group, 26

T123 Leg numbering notation for tensor products, 12τ Finite trace on a von Neumann algebra, 23T Map T : H → K, 59 Algebraic tensor product, 11⊗ Tensor product of von Neumann algebras, 13⊗N Connes’ tensor product, 53T (G) Trigonometric polynomials on the compact group G,

120Tr Non-normalized trace on a matrix algebra, 11tr Normalized trace on a matrix algebra, 11

U(H) Group of unitary elements in B(H), 18

Vπ Eigenmatrices satisfying (id⊗σg)(Vπ) = Vπ(π(g)⊗1),131

ξ∗ Element of H viewed as ξ∗ : H → C, 11ξ Elements of H, 58

Z2(Γ, S1) S1-valued 2-cocycles on Γ, 34Z(M) Center of M , 21

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