Topics in K-theory of operator algebras:Dadarlat-Pennig’s generalization of Dixmier-Douadytheory and group actions on Kirchberg algebras I, II

泉　正己

京都大学　大学院理学研究科

2018 年 9 月 12, 13 日, 信州大学

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C∗-algebras

H: Hilbert space.B(H): the set of bounded operators on H.

B(H) is an algebra over C.B(H) is a Banach space w.r.t. operator norm

∥T∥ = supξ∈H\0

∥Tξ∥∥ξ∥

.

B(H) has a ∗-operation T 7→ T ∗, where ⟨Tξ, η⟩ = ⟨ξ, T ∗η⟩, ∀ξ, η ∈ H.

Definition

A C∗-algebra is a subalgebra of B(H) closed under the norm topology and the∗-operation.

Theorem (Gelfand-Naimark)

A Banach ∗-algebra A satisfying the C∗-condition ∥T ∗T∥ = ∥T∥2 for any T ∈ Ais isomorphic to a C∗-algebra.

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Examples of C∗-algebras

Example

B(H), Mn = B(Cn).

C(X) with ∥f∥ = maxx∈X |f(x)| and f∗(x) = f(x).Here C(X) is the set of continuous functions on a compact Hausdorff spaceX.

In what follows,H is a separable infinite dimensional Hilbert space, andX is a compact metric space (often a finite CW-complex).

Example

A: C∗-algebra,The set of continuous A-valued functions on X, denoted by C(X,A), is aC∗-algebra with pointwise operations and ∥f∥ = maxx∈X ∥f(x)∥.

C(X,A) ∼= C(X)⊗A.

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(Locally trivial) Continuous fields of C∗-algebras

Let A be a C∗-algebra.

For a fiber bundle π : E → X with a fiber A and structure group Aut(A), the setof continuous sections Γ(E) is a C∗-algebra with fiberwise operations and norm∥s∥ = maxx∈X ∥s(x)∥.

We call it a locally trivial continuous field of A over X.

Example

C(X,A) = Γ(E) for a trivial bundle E.

Whenever we discuss isomorphisms between (locally trivial) continuous fields overX, we assume that they are C(X)-module maps.

We denote by BunX(A) the set of isomorphism classes of locally trivial fields ofA over X.

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The C∗-algebra K

The set of compact operators K on H is a C∗-algebra.

DefinitionA C∗-algebra A is said to be stable if A is isomorphic to A⊗K.C∗-algebras A and B are stably isomorphic if A⊗K ∼= B ⊗K.

Since H ⊗H ∼= H, we have K⊗K ∼= K, and A⊗K is always stable for anyC∗-algebra A.

We have a short exact sequence

0 → T → U(H) → Aut(K) → 0, (1)

with U(H) ∋ U 7→ AdU ∈ Aut(K), where U(H) is the set of unitary operatorson H equipped with the strong operator topology, which is contractible.

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Dixmier-Douady theorem

Let A = Γ(E), B = Γ(E′) be locally trivial continuous fields of K over X.

Then A⊗C(X) B = Γ(E ⊗E′), which is also a locally trivial continuous field of Kover X as K⊗K ∼= K.

Theorem (Dixmier-Douady 1963)

The set BunX(K) of isomorphism classes of locally trivial fields of K over Xforms an abelian group isomorphic to H3(X;Z) under operation of tensor productover C(X).

The Dixmier-Douady class δ(A) ∈ H3(X;Z) for A = Γ(E) is a characteristicclass of the fiber bundle p : E → X with fiber K.

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Homotopy theory approach

Recall that T is an Eilenberg-MacLane space of type K(Z, 1), i.e.

πi(T) =

0, i = 1Z, i = 1

.

Since U(H) is contractible, U(H) → Aut(K) ∼= U(H)/T is a universal principalT-bundle and Aut(K) is a classifying space of T, i.e. U(H) = ET,Aut(K) = BT, and in particular Aut(K) is a K(Z, 2) space.

The classifying space BAut(K) is a K(Z, 3) space, and

BunX(K) ∼= [X,BAut(K)] = [X,K(Z, 3)] ∼= H3(X;Z).

Goal of the talkFor any strongly self-absorbing C∗-algebra D, the homotopy set[X,BAut(D ⊗K)] gives the first group E1

D(X) of a generalized cohomologytheory E∗

D(X) (Dadarlat-Pennig 2016).

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K0-groups

For a C∗-algebra A, set P(A) := p ∈ A; p = p2 = p∗.

Recall that p ∈ P(C(X,K)) = P(C(X)⊗K) gives a vector bundle:⊔x∈X

p(x)H → X.

Definition (Murray-von Neumann equivalence)

For p, q ∈ A, we set p ∼ q if ∃v ∈ A s.t. v∗v = p and vv∗ = q.

Since K⊗M2 = K(H ⊗ C2) ∼= K, the set P(A⊗K)/ ∼ is a semigroup with

p⊕ q =

(p 00 q

)∈ P(A⊗K⊗M2) ∼= P(A⊗K).

Definition (Unital case)

K0(A) is the Grothendieck group of P(A⊗K)/ ∼.We denote by K0(A)+ the image of P(A⊗K)/ ∼ in K0(A).

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K1-groups

For a unital C∗-algebra A, set U(A) := u ∈ A; u∗u = uu∗ = 1, andUn(A) = U(A⊗Mn).

Recall K1(X) = [X,U(n)] for sufficiently large n.Since Map(X,U(n)) = U(C(X,Mn)) = U(C(X)⊗Mn) = Un(C(X)),we have [X,U(n)] = Un(C(X))/Un(C(X))0.

Definition

K1(A) is the inductive limit lim−→

Un(A)/Un(A)0 with a connection map

Un(A) ∋ u 7→ u⊕ 1 ∈ Un+1(A).

K∗(C(X)) = K∗(X).K∗(A) = K∗(A⊗Mn) = K∗(A⊗K).

Example

For A = Mn, (K0(A),K0(A)+, [1]0,K1(A)) = (Z,Z+, n, 0).

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Non-unital case

For a based space (X,x0), denote C(X,x0) = f ∈ C(X); f(x0) = 0.

Recall Ki(X) = ker(ι∗ : Ki(X) → Ki(x0)).We need a definition of Ki(A) satisfying Ki(C(X,x0)) = Ki(X).

Definition (Non-unital case)

For a non-unital C∗-algebra A, set Ki(A) := ker(π∗ : Ki(A+ C1) → Ki(C)).

For a locally compact Hausdorff space Y , set C0(Y ) = C(Y ∪ ∞,∞).The suspension of A is SA := A⊗ C0(0, 1) ∼= A⊗ C0(R).

SC(X,x0) = C(ΣX,Σx0) where ΣX = (X × I)/(x0 × I ∪X × ∂I).

Theorem (Bott periodicity)

K0(SA) ∼= K1(A) and K1(SA) ∼= K0(A).

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Twisted K-theory

Recall K∗(C(X,K)) = K∗(X).

What happens if the trivial continuous field C(X,K) of K over X is replaced witha locally trivial one?

Definition (J. Rosenberg 1988)

The twisted K-theory Kτ+∗(X) with τ ∈ H3(X,Z) is defined by K∗(A), whereA is a locally trivial continuous field of K over X whose Dixmier-Douady classδ(A) is τ (cf. Donovan-Karoubi 1970).

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Kasparov KK-theory

KK(A,B) is a bivariant functor from the category of (separable) C∗-algebras tothe category of abelian groups satisfying

∃ associative product KK(A,B)×KK(B,C) → KK(A,C).

KK(C, B) = K0(B).

KK(A,C) = K0(A), where K0(C(X)) = K0(X).

If A is in a bootstrap category N of Rosenberg-Schochet, the UCT sequence

0 → Ext(K(A),K(B)) → KK(A,B) → Hom(K(A),K(B)) → 0

is exact.

If A and B are Kirchberg algebras,

KK(A,B) ∼= Hom(A⊗K, B ⊗K)/ ∼

where ρ ∼ σ iff ∃utt∈[0,∞) in U(B ⊗K+ C1) s.t. limt→∞

utρ(a)u∗t = σ(a).

ρ and σ as above is said to be asymptotically unitarily equivalent.

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KK-equivalence

Definition

A and B are KK-equivalent if ∃x ∈ KK(A,B), ∃y ∈ KK(B,A) s.t.x⊗y = KK(idA) ∈ KK(A,A) and y⊗x = KK(idB) ∈ KK(B,B).

A, A⊗Mn, and A⊗K are mutually KK-equivalent.

C and S2C = C0(R2) = C0(C) are KK-equivalent.Define pB ∈ P((C0(C) + C1)⊗M2) = P(C(CP 1)⊗M2) by

pB(z) =1

1 + |z2|

(|z|2 zz 1

).

Then x = [pB ]0 − [1]0 ∈ K0(C0(R2)) = KK(C, C0(R2)) is an invertible element.

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The UHF algebras

For a sequence nk∞k=1 of integers greater than 1, we set

Am = Mn1 ⊗Mn2 ⊗ · · · ⊗Mnm∼= Mn, n =

m∏k=1

nk.

Embedding Am into Am+1 by a 7→ a⊗ 1Mnm+1, we get an inductive system of

C∗-algebras Am∞m=1.

The norm completion of its inductive limit is a C∗-algebra, called the UHFalgebra, and denoted by

⊗∞k=1 Mnk

.

A UHF algebra A is said to be of infinite type if A⊗A ∼= A, e.g.Mn∞ =

⊗∞k=1 Mn,

MQ =⊗∞

k=1 Mk!. The universal UHF algebra.

UHF algebras of infinite type are strongly self-absorbing.

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The classification of UHF algebras

UHF algebras are completely classified by the supernatural number

∞∏k

nk =∏

p:prime

pap , ap = 0, 1, 2, . . . ,∞,

or more precisely, the set app is a complete invariant (Glimm 1960).

Definition

For a C∗-algebra A, a linear functional τ : A→ C is a trace if ∥τ∥ = τ(1) = 1 andτ(xy) = τ(yx) for ∀x, y ∈ A.

The normalized trace τm = 1n1n2···nm

Tr on⊗m

k=1 Mnkextends to a unique trace

τ on⊗∞

k=1 Mnk, and

τ(p); p ∈ P(∞⊗k=1

Mnk) = (

∞∪m=1

1

n1n2 · · ·nmZ) ∩ [0, 1].

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Elliott Theorem

An inductive limit of finite dimensional C∗-algebras is said to be AF algebra.

Theorem (Elliott 1976)

The isomorphism class of an AF algebras A is completely determined by(K0(A),K0(A)+, [1]0).

Since K1(Mn) = 0, the K1-group is trivial for any AF algebra.

For A =⊗∞

k=1 Mnk,

(K0(A),K0(A)+, [1]0) = (

∞∪m=1

1

n1n2 · · ·nmZ,K0(A) ∩ R+, 1).

K0(Mn∞) = Z[ 1n ].K0(MQ) = Q.They have ring structure.

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The Cuntz algebras O2 and O∞

The Cuntz algebra On for n ≥ 2 is the universal C∗-algebra with generatorsSini=1 and relations

S∗i Sj = δi,j1,

n∑i=1

SiS∗i = 1.

For n = ∞, we define the Cuntz algebra O∞ by imposing only the first relation.

Their K-groups are

K0(On) =

Z/(n− 1)Z, n <∞Z, n = ∞ , K1(On) = 0.

K∗(X;Zp) ∼= K∗(C(X)⊗Op+1).

The Cuntz algebras O2 and O∞ are strongly self-absorbing.

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Kirchberg-Phillips Theorem

The Cuntz algebras are typical examples of Kirchberg algebras, separable nuclearsimple purely infinite C∗-algebras.

Theorem (Phillips 2000)

Any two Kirchberg algebras A and B are stably isomorphic iff they areKK-equivalent.If moreover they are in the bootstrap class N of Rosenberg-Schochet, they areisomorphic iff

(K0(A), [1A]0,K1(A)) ∼= (K0(B), [1B ]0,K1(B)).

O2 is KK-equivalent to 0, and it plays the role of 0, i.e. O2 ⊗A ∼= O2 for anyunital simple separable nuclear C∗-algebra A.

O∞ is KK-equivalent to C, and it plays the role of 1, i.e. O∞ ⊗B ∼= B for anyKirchberg algebra B.

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Jiang-Su algebra Z

The Jiang-Su algebra Z is a mysterious simple stably finite C∗-algebra withouthaving projections other than 0 and 1.

Z is KK-equivalent to C, having K0(Z) = Z and K1(Z) = 0.

Z is absorbed by every strongly self-absorbing C∗-algebra by tensor product.

In fact, whether a given C∗-algebra absorbs Z or not is the most importantcriterion of classifiability of it.

Z is an inductive limit of building blocks Ip,q where p and q are mutually primenatural numbers and

Ip,q = f ∈ C([0, 1],Mp ⊗Mq); f(0) ∈ Mp ⊗ C1, f(0) ∈ C1⊗Mq.

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Strongly self-absorbing C∗-algebras

The notion of strongly self-absorbing C∗-algebras was introduced by Toms-Winterin 2007 to single out the class of C∗-algebras playing distinguished roles in theclassification of nuclear C∗-algebras.

Definition (Toms-Winter, 2007)

A C∗-algebra D with unit is said to be strongly self-absorbing if there exist anisomorphism ψ : D → D ⊗D and a sequence of unitaries Un∞n=1 ⊂ U(D ⊗D)such that for any T ∈ D, and we have

limn→∞

Un(T ⊗ 1)U∗n = ψ(T ).

The sequence Un∞n=1 can be replaced by a continuous family U(t)t∈[0,∞)

with U(0) = 1, and ψ is arbitrary once it exists.

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Basic properties

Theorem

Let D be a strongly self-absorbing C∗-algebra.

(1) D is simple nuclear either stably finite or purely infinite; if it is stably finite,then it admits a unique trace.

(2) K0(D) has a ring structure with unit [1] given by[p][q] = [ψ−1(p⊗ q)] ∈ K0(A) for any projections p, q ∈ D.

(3) K1(D) = 0 if D is in the bootstrap class N of Rosenberg-Schochet.

(4) Aut(D) is contractible.

(5) Aut0(D ⊗K) has homotopy type of a CW-complex.

All the known strongly self-absorbing C∗-algebras are in the following list:C, the Cuntz algebras O2 and O∞, the UHF algebras of infinite type,the Jiang-Su algebra Z, and the tensor product of the Cuntz algebra O∞ and theUHF algebras of infinite type.

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Proof of (4) of Aut(D)

Lemma

If there exists a continuous path ψt∈[0,1] in Hom(D,D ⊗D) satisfyingψ0(x) = x⊗ 1, ψ1(x) = 1⊗ x, and ψt is an isomorphism for any 0 < t < 1, thenAut(D) is contractible.

Proof.

For α ∈ Aut(D), set

H(α, t) =

α, t = 0,ψ−1t (α⊗ id) ψt, 0 < t < 1,

id, t = 1.

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Infinite loop space

For a pointed topological space E, we denote by ΩE its loop space i.e.

ΩE = f ∈ Map([0, 1], E); f(0) = f(1) = ∗.

DefinitionA pointed topological space E = E0 is said to be an infinite loop space if thereexists a sequence of pointed topological spaces En∞n=1 such that ΩEn ishomotopy equivalent to En−1.Such a sequence is called an Ω-spectrum.

A Ω-spectrum En∞n=0 gives rise to a generalized cohomology via the homotopysets [X,En].

Example

En = K(A,n), [X,En] = Hn(X;A).

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Dadarlat-Pennig theorem

Theorem (Dadarlat-Pennig, 2016)

Let X be a compact metrizable space, and let D be a strongly self-absorbingC∗-algebra.The set BunX(D ⊗K) of isomorphism classes of locally trivial fields of D ⊗Kover X is an abelian group under operation of tensor product over C(X).Moreover, the group is isomorphic to E1

D(X), the first group of a generalizedconnective cohomology theory E∗

D(X) defined by the infinite loop spaceBAut(D ⊗K).

There exists an Ω-spectrum En with E0 = Aut(D ⊗K), E1 = BAut(D ⊗K),and En

D(X) = [X,En].

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Symmetric monoidal category B⊗

Let B⊗ be a topological category whose objects are Z+ (n is identified with(D ⊗K)⊗n) and morphsims are

Hom(0, n) = p ∈ P((D ⊗K)⊗n); [p]0 ∈ K0(D⊗n)×+,

Hom(m,n) = ρ ∈ Hom((D ⊗K)⊗m), (D ⊗K)⊗n); KK(ρ) is invertible,

for m ≥ 1.

B⊗ is a symmetric monoidal category with m⊗ n = m+ n.

Note Aut(D ⊗K) ⊂ Hom(1, 1) = ρ ∈ End(D ⊗K);KK(ρ) is invertible.

BAut(D ⊗K) ∼= BB⊗.

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Coefficients

The E2-page of the Atiyah-Hirzebruch spectral sequence for E∗D(X) is given by

Ep,q2 = Hp(X,Eq

D(pt)) = Hp(X,π−q(Aut(D ⊗K))).

Theorem (Dadarlat-Pennig, 2016)

Let D be a strongly self-absorbing C∗-algebra not isomorphic to C satisfying theUCT. Then

π2i(Aut(D ⊗K)) ∼=K0(D)×+, i = 0K0(D), i ≥ 1

,

π2i−1(Aut(D ⊗K)) ∼= 0.

Recall that K0(D) has a ring structure, which also has a positive cone K0(D)+generated by the classes represented by projections.

O2 O∞ O∞ ⊗Mn∞ O∞ ⊗MQ Mn∞ MQ ZK0(D) 0 Z Z[ 1n ] Q Z[ 1n ] Q ZK0(D)×+ 0 1,−1 Z[ 1n ]

× Q× Z[ 1n ]×+ Q×

+ 1

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Examples

Since the differentials of the Atiyah-Hirzebruch spectral sequence are known to betorsion operators, we get

Corollary

(1) For a connected compact metrizable space X,

BunX(MQ ⊗K) ∼= H1(X,Q×+)⊕

⊕k≥1

H2k+1(X,Q),

BunX(O∞ ⊗MQ ⊗K) ∼= H1(X,Q×)⊕⊕k≥1

H2k+1(X,Q).

(2) For a connected finite CW-complex X with H∗(X,Z) torsion free,

BunX(Z ⊗K) ∼=⊕k≥1

H2k+1(X,Z),

BunX(O∞ ⊗K) ∼= H1(X,Z/2Z)⊕⊕k≥1

H2k+1(X,Z).

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[X,Aut(D ⊗K)]

TheoremAssume X is connected and D is a strongly self-absorbing. Then

[X,Aut(D ⊗K)] ∼= K0(C(X)⊗D)×+.

[X,Aut(Z ⊗K)] ∼= (1 + K0(X))×,

[X,Aut(O∞ ⊗K)] ∼= (±1 + K0(X))×.

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[X,Aut(A)] for Kirchberg algebras

Theorem (Dadarlat 2007)

For a uital Kirchberg algebra A, there are bijection

χ : [X,Aut(A)0] → KK(CνA,SC(X,x0)⊗A),

χ : [X,Aut(A⊗K)] → KK(A,C(X,x0)⊗A),

where CνA = f ∈ C([0, 1], A); f(0) ∈ C1A, f(1) = 0 is the mapping cone ofthe inclusion map ν : C → A.If moreover (X,x0) is a H

′-space, they are group isomorphisms.

The exact sequence 0 → SA→ CνA→ C → 0 implies the exact sequence

KK(A,SC(X,x0)⊗A)ν∗

−→ K1(C(X,x0)⊗A) → KK(CνA,SC(X,x0)⊗A)

→ KK(A,C(X,x0)⊗A)ν∗

−→ K0(C(X,x0)⊗A) → · · ·

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[X,Aut(On+1)]

For the Cuntz algebra On+1, we have [X,Aut(On+1)] ∼= K1(X;Zn) as sets.

Recall the universal coefficient exact sequence:

0 → K1(X)⊗ Zn → K(X;Zn)δ−→ Tor(K0(X),Zn) → 0.

Theorem (I, 2018)

[X,Aut(On+1)] is isomorphic to K1(X;Zn) equipped with group structure

x y = x+ y − xδ(y).

In particular, [X,Aut(On+1)] is a group extension

0 → K1(X)⊗ Zn → [X,Aut(On+1)] → (1 + Tor(K0(X),Zn))× → 0.

Since y−1xy = x(1− δ(y)) for x ∈ K1(X;Zn), y ∈ K1(X)⊗ Zn, if

(K1(X)⊗ Zn)Tor(K0(X),Zn)) = 0, the group [X,Aut(On+1)] is

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[X,BAut(On+1)]

Theorem (Dadarlat 2012+Sogabe-I. 2018)

If H∗(X;Z) has no n-torsion, #[X,BAut(On+1)] = #K0(X)⊗ Zn.

The canonical map U(n+ 1) → Aut(On+1) induces a surjetion[X,BU(n+ 1)] → [X,BAut(On+1)] if n is sufficiently large compared to dimX.

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Group actions

Recall for two discrete groups Γ, Λ, the classifying map

Hom(Γ,Λ)/conjugacy ∋ [φ] 7→ [Bφ] ∈ [BΓ, BΛ]

is a bijection.

We try to classify group actions on C∗-algebra follwing the same spirit.

Let G be a discrete group and let A be a C∗-algebra.A G-action on A is a homomorphism from G to Aut(A).

Two G-actions α, β are coycle conjugate if ∃θ ∈ Aut(A) and∃ugg∈G ⊂ U(A) satisfying ugh = ugαg(uh) and Adug αg = θ βg θ−1.

A G-action α is outer if αg is outer for any g ∈ G \ e.

Denote by OA(G,A) the set of outer actions of G on A.

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Conjecture

Conjecture

Let G be a discrete torsion-free amenable group, and let A be a stable Kirchbergalgebra. Then the map

OA(G,A)/cocycle conjugacy ∋ [α] 7→ [Bαs] ∈ [BG,BAut(A⊗K)]

is a bijection, where αsg = αg ⊗Ad ρg acts on A⊗K(ℓ2(G)) and ρ is the right

regular representation.

Theorem (Matui-I. 2018)

The conjecture is true for any poly-Z group G.

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Poly-Z group actions

A group G is poly-Z if there exists a subnormal series

e = G0 ≤ G1 ≤ · · · ≤ Gn = G,

satisfying Gk/Gk−1∼= Z for any k = 1, 2, . . . , n.

The number n is called the Hirsch length of G, denoted by h(G), which coincideswith the cohomological dimension of G.

In fact, there exists a free cocompact polynomial action of G on Rn, and we canchoose EG = Rn and BG = Rn/G.

Example

Cocycle conjugacy classes of outer Z3-actions on O∞ ⊗K are in one-to-onecorrespondence with

BunT3(O∞ ⊗K) ∼= H1(T3;Z/2Z)⊕H3(T3;Z) ∼= (Z/2Z)3 ⊕ Z.

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Unital case

For a G-action α on A, we define a principal Aut(A)-bundle Pα over BG by

Pα := (EG×Aut(A))/G→ BG,

where g · (x, γ) = (g · x, αg γ).

Theorem (Matui-I. 2018)

Let G be a poly-Z group and let A be a unital Kirchberg algebra.Then α, β ∈ OA(G,A) are cocycle conjugate iff Pαs ∼= Pβs .

Example

Cocycle conjugacy classes of outer Z3-actions on On are in one-to-onecorrespondence with

H2(T3;π1(Aut(On ⊗K))) = H2(T3;Zn−1) ∼= (Z/2(n− 1)Z)3.

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