tensor norms in operator algebras

Wydział Matematyki i InformatykiUniwersytetu im. Adama Mickiewicza w Poznaniu

Środowiskowe Studia Doktoranckiez Nauk Matematycznych

Tensor Norms in OperatorAlgebras

Marius Junge

University of Illinois, [email protected]

Publikacja współfinansowana ze środków Uni Europejskiejw ramach Europejskiego Funduszu Społecznego

M. Junge, Tensor Norms in Operator Algebras

Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction to tensor norms, basic properties . . . . . . . . . . . . . . . . . . . . . 3

Some tensor norms on operator algebras . . . . . . . . . . . . . . . . . . . . . . . . . 12

Completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Completely bounded maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Operator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Operator space tensor norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Local reflexivity and exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Kirchberg theory I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Joint probabilities and quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . 71

Matrix valued Tsirelson’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Violation for tripartite correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Problems for grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Marius Junge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

References

[1] Archbold, R. J.; Batty, C. J. K.: C∗-tensor norms and slice maps. J. London Math. Soc. (2)22 (1980), no. 1, 127–138.

[2] Blecher, David P.; Paulsen, Vern I.: Tensor products of operator spaces. J. Funct. Anal. 99(1991), no. 2, 262–292.

[3] Brown, Nathanial P.; Ozawa, Narutaka: C∗-algebras and finite-dimensional approximations.Graduate Studies in Mathematics, 88. American Mathematical Society, Providence, RI, 2008.xvi+509 pp. ISBN: 978-0-8218-4381-9; 0-8218-4381-8

[4] Defant, Andreas; Floret, Klaus: Tensor norms and operator ideals. North-Holland Mathema-tics Studies, 176. North-Holland Publishing Co., Amsterdam, 1993. xii+566 pp

[5] Effros, Edward G.; Lance, E. Christopher: Tensor products of operator algebras. Adv. Math.25 (1977), no. 1, 1–34. 12

[6] Junge, M.; Navascues, M.; Palazuelos, C.; Perez-Garcıa, D.; Scholz, V. B.; Werner, R.F.(D-HANN-TP) Connes embedding problem and Tsirelson’s problem. J. Math. Phys. 52(2011), no. 1, 012102, 12 pp.

[7] Kirchberg, Eberhard: On nonsemisplit extensions, tensor products and exactness of groupC∗-algebras. Invent. Math. 112 (1993), no. 3,449–489. 57

[8] Ozawa, Narutaka: About the QWEP conjecture. Internat. J. Math. 15 (2004), no. 5,501–53033, 57

[9] Paulsen, Vern I. Completely bounded maps and dilations. Pitman Research Notes in Mathe-matics Series, 146. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., NewYork, 1986. xii+187 pp. ISBN: 0-582-98896-9 12, 19, 29

[10] Perez-Garcıa, D.; Wolf, M. M.; Palazuelos, C.; Villanueva, I.; Junge, M.: Unbounded violationof tripartite Bell inequalities. Comm. Math. Phys. 279 (2008), no. 2, 455–486. 82

1


[11] Pisier, Gilles: Introduction to operator space theory. London Mathematical Society Lectu-re Note Series, 294. Cambridge University Press, Cambridge, 2003. viii+478 pp. ISBN:0-521-81165-1 29, 33

[12] Pisier, Gilles: Grothendieck’s Theorem, past and present. arXiv:1101.4195v3 [math.FA]

C∗-algebras and Functional analysis

[13] Conway, John B.: A course in functional analysis. Second edition. Graduate Texts in Mathe-matics, 96. Springer-Verlag, New York, 1990. xvi+399 pp. ISBN: 0-387-97245-5 3

[14] Kadison, Richard V.; Ringrose, John R.: Fundamentals of the theory of operator algebras. Vol.II. Advanced theory. Corrected reprint of the 1986 original. Graduate Studies in Mathematics,16. American Mathematical Society, Providence, RI, 1997. pp. i–xxii and 399–1074. ISBN:0-8218-0820-6 3

[15] Takesaki, M.: Theory of operator algebras. I. Reprint of the first (1979) edition. Encyclopa-edia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 5.Springer-Verlag, Berlin, 2002. xx+415 pp. ISBN: 3-540-42248. 3

2


Introduction to tensor norms, basic properties(following [13, 14, 15]

3

1/8 Tensor norms (2/10)2012-05-14 06:45:57

2/8 Tensor norms (3/10)2012-05-14 06:45:57

3/8 Tensor norms (4/10)2012-05-14 06:45:58

4/8 Tensor norms (5/10)2012-05-14 06:45:58

5/8 Tensor norms (6/10)2012-05-14 06:45:58

6/8 Tensor norms (7/10)2012-05-14 06:45:58

7/8 Tensor norms (8/10)2012-05-14 06:45:58

8/8 Tensor norms (9/10)2012-05-14 06:45:58


Some tensor norms on operator algebras(following Effros Lance [5] and [9])

Key Words: State space of a tensor product, minimal norm, maximal norm

12

1/6 Operator algebras (1/24)2012-05-15 09:21:59


Completely positive maps(following Paulsen [9]

Key Words: Operator systems, Stinespring’s GNS construction, extension theorem

19


Completely bounded maps(follwing [9] and [11])

Key Words: Operator spaces, completely bounded maps, Paulsen system, independence

in the definition of min tensor product

29


Operator spaces(see [11] and [8])

33


Basic properties

34

1/5 Operator spaces (1/21)2012-05-15 11:46:35

2/5 Operator spaces (2/21)2012-05-15 11:46:35

3/5 Operator spaces (3/21)2012-05-15 11:46:36

4/5 Operator spaces (4/21)2012-05-15 11:46:36

5/5 Operator spaces (5/21)2012-05-15 11:46:36


Operator space tensor norms

40

1/3 Operator spaces (6/21)2012-05-15 11:55:36

2/3 Operator spaces (7/21)2012-05-15 11:55:36

3/3 Operator spaces (8/21)2012-05-15 11:55:36


Local reflexivity and exactness

44

1/12 Operator spaces (9/21)2012-05-15 11:56:41

2/12 Operator spaces (10/21)2012-05-15 11:56:41

3/12 Operator spaces (11/21)2012-05-15 11:56:42

4/12 Operator spaces (12/21)2012-05-15 11:56:42

5/12 Operator spaces (13/21)2012-05-15 11:56:42

6/12 Operator spaces (14/21)2012-05-15 11:56:42

7/12 Operator spaces (15/21)2012-05-15 11:56:42

8/12 Operator spaces (16/21)2012-05-15 11:56:43

9/12 Operator spaces (17/21)2012-05-15 11:56:43

10/12 Operator spaces (18/21)2012-05-15 11:56:43

11/12 Operator spaces (19/21)2012-05-15 11:56:43

12/12 Operator spaces (20/21)2012-05-15 11:56:43


Kirchberg theory I(see [7, 8])

Key Words: Full C* of the free group, WEP and QWEP, and LLP

57

1/13 Kirchberg theory (1/13)2012-05-15 12:39:08


Joint probabilities and quantum mechanics

Key Words: Grothendieck’s inequality, Tsirelson’s reformulation, real version and

Clifford algebras

71

1/3 Violation (1/10)2012-05-21 11:06:06

2/3 Violation (2/10)2012-05-21 11:06:06

3/3 Violation (3/10)2012-05-21 11:06:07

1/1 Tsirelson (1/8)2012-05-21 11:07:55


Matrix valued Tsirelson’s problem(follwing 6)

76

1/5 Tsirelson (1/5)2012-05-15 19:25:11

2/5 Tsirelson (2/5)2012-05-15 19:25:11

3/5 Tsirelson (3/5)2012-05-15 19:25:12

4/5 Tsirelson (4/5)2012-05-15 19:25:12

5/5 Tsirelson (5/5)2012-05-15 19:25:12


Violation for tripartite correlation(following original article [10] and Briet/Viddick arXiv:1108.5647)

82

1/6 Violation (5/10)2012-05-21 11:09:20

2/6 Violation (6/10)2012-05-21 11:09:20

3/6 Violation (7/10)2012-05-21 11:09:20

4/6 Violation (8/10)2012-05-21 11:09:20

5/6 Violation (9/10)2012-05-21 11:09:20

6/6 Violation (10/10)2012-05-21 11:09:21


Problems for grades

89

Problems

(1) What exactly is an integral operator in Banach space theory and how does it relate

to (X ⊗ε Y )∗ (see Pietsch book on operator ideals (s-numbers) and/or Defant-

Floret)

(2) Recall the definition of the operator space projective tensor product and show that

in fact

(X⊗Y )∗ ∼= CB(X, Y ∗) .

(3) Show that (F ⊗min K)∗ = F ∗⊗S1, the operator space projective tensor product.

(4) Show that the operator spaces E1n generated by 1, λ(g1), ..., λ(gn−1) and E2

n gener-

ated by λ(g1), ..., λ(gn) in C∗Fn−1 and C∗Fn are completely isometric. Repeat the

exercise for the reduced C∗-algebra.

(5) Show that min tensor product commutes with direct limits. Can you show the

same for the max tensor product? What are the correct assumptions (Hint see

Paulsen’s work on tensor product of operator systems).

(6) Show that `n1 embeds completely isometrically on ∗ni=1`2∞. (Hint: First show that

‖n∑

k=1

ak ⊗ ek‖ = supukunitaryuk=u−1

k

‖∑

k

ak ⊗ uk‖ .

holds for all matrices.)

(7) Find out what Pisier/Shlyaktenko’s operator space Grothendieck inequality is and

use this to prove

ex(Sn1 ) ∼ n .

(The upper estimate was given in the lecture).

(8) Let A, B be C∗-algebras. Show that the inclusion A ⊗max B ⊂ A∗∗ ⊗max B∗∗ is

isometric.

(9) Show that A∗∗ semidiscrete implies A∗∗ injective. Also show that for a von Nueu-

mann algebra N

N∗⊗X ⊂ N∗⊗Y isometrically

whenever X ⊂ Y (completely isometrically) implies that A is injective.

(10) (Pisier p315). Show that for an C∗-algebra A, the implication

A∗∗ injective ⇒ A locally reflexive

Conclude that nuclear C∗-algebras are locally reflexive. Deduce with the help of

Kirchberg’s hard theorem (exact=subnuclear) that exact C∗-algebras are locally

reflexive.

1


Marius Junge

Prof. Junge is a specialist in Functonal Analysis, C∗-algebras, noncommutative Lpspaces, Quantum Information Theory.

Diploma in Mathematics, Ph. D and Habilitation in Christian-Albrechts-Universitat in

Kiel (1989, 1991, 1996) under supervision of prof. H. Konig. Till 1999 he worked at Kiel,

then in Odense (Denmark) and from 1999 at University of Illinois, Urbana-Champaign

(from 2007 full professor).

Prof. Junge is a member of the Editorial Board of Proc. AMS and Illinois Journal of

Mathematics. He held visiting positions at IHP, Univ. Besancon and Paris. Author of 67

publications, among others in Inventiones, Journal AMS, Annals of Mathematics.

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tensor norms in operator algebras

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