tensor norms in operator algebras
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Wydział Matematyki i InformatykiUniwersytetu im. Adama Mickiewicza w Poznaniu
Środowiskowe Studia Doktoranckiez Nauk Matematycznych
Tensor Norms in OperatorAlgebras
Marius Junge
University of Illinois, Urbana-Champaignjunge@math.uiuc.edu
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
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M. Junge, Tensor Norms in Operator Algebras
[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
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M. Junge, Tensor Norms in Operator Algebras
Introduction to tensor norms, basic properties(following [13, 14, 15]
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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
M. Junge, Tensor Norms in Operator Algebras
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
2/6 Operator algebras (2/24)2012-05-15 09:22:00
3/6 Operator algebras (3/24)2012-05-15 09:22:00
4/6 Operator algebras (4/24)2012-05-15 09:22:00
5/6 Operator algebras (5/24)2012-05-15 09:22:01
6/6 Operator algebras (6/24)2012-05-15 09:22:01
M. Junge, Tensor Norms in Operator Algebras
Completely positive maps(following Paulsen [9]
Key Words: Operator systems, Stinespring’s GNS construction, extension theorem
19
1/9 Operator algebras (7/26)2012-05-16 09:24:12
2/9 Operator algebras (8/26)2012-05-16 09:24:13
3/9 Operator algebras (9/26)2012-05-16 09:24:13
4/9 Operator algebras (10/26)2012-05-16 09:24:13
5/9 Operator algebras (11/26)2012-05-16 09:24:13
6/9 Operator algebras (12/26)2012-05-16 09:24:13
7/9 Operator algebras (13/26)2012-05-16 09:24:14
8/9 Operator algebras (14/26)2012-05-16 09:24:14
9/9 Operator algebras (15/26)2012-05-16 09:24:14
M. Junge, Tensor Norms in Operator Algebras
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
1/3 Operator algebras (10/19)2012-05-10 08:02:42
2/3 Operator algebras (11/19)2012-05-10 08:02:42
3/3 Operator algebras (12/19)2012-05-10 08:02:42
M. Junge, Tensor Norms in Operator Algebras
Operator spaces(see [11] and [8])
33
M. Junge, Tensor Norms in Operator Algebras
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
M. Junge, Tensor Norms in Operator Algebras
Operator space tensor norms
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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
M. Junge, Tensor Norms in Operator Algebras
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
M. Junge, Tensor Norms in Operator Algebras
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
2/13 Kirchberg theory (2/13)2012-05-15 12:39:08
3/13 Kirchberg theory (3/13)2012-05-15 12:39:08
4/13 Kirchberg theory (4/13)2012-05-15 12:39:09
5/13 Kirchberg theory (5/13)2012-05-15 12:39:09
6/13 Kirchberg theory (6/13)2012-05-15 12:39:09
7/13 Kirchberg theory (7/13)2012-05-15 12:39:09
8/13 Kirchberg theory (8/13)2012-05-15 12:39:09
9/13 Kirchberg theory (9/13)2012-05-15 12:39:10
10/13 Kirchberg theory (10/13)2012-05-15 12:39:10
11/13 Kirchberg theory (11/13)2012-05-15 12:39:10
12/13 Kirchberg theory (12/13)2012-05-15 12:39:10
13/13 Kirchberg theory (13/13)2012-05-15 12:39:10
M. Junge, Tensor Norms in Operator Algebras
Joint probabilities and quantum mechanics
Key Words: Grothendieck’s inequality, Tsirelson’s reformulation, real version and
Clifford algebras
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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
M. Junge, Tensor Norms in Operator Algebras
Matrix valued Tsirelson’s problem(follwing 6)
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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
M. Junge, Tensor Norms in Operator Algebras
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
M. Junge, Tensor Norms in Operator Algebras
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.
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M. Junge, Tensor Norms in Operator Algebras
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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