Tensor Products of C*-Algebras and Operator Spaces
The Connes–Kirchberg Problem

Presents an important open problem on operator algebras in a style accessible to young researchers or Ph.D. students.

Gilles Pisier (Author)

9781108749114, Cambridge University Press

Paperback / softback, published 27 February 2020

494 pages
22.7 x 15.3 x 2.7 cm, 0.7 kg

'This book is jam packed with information, and should be an invaluable guide to anyone interested in these ideas … For the complete picture and the recent advances, Pisier's book is the place to go.' Bulletin of the American Mathematical Society

Based on the author's university lecture courses, this book presents the many facets of one of the most important open problems in operator algebra theory. Central to this book is the proof of the equivalence of the various forms of the problem, including forms involving C*-algebra tensor products and free groups, ultraproducts of von Neumann algebras, and quantum information theory. The reader is guided through a number of results (some of them previously unpublished) revolving around tensor products of C*-algebras and operator spaces, which are reminiscent of Grothendieck's famous Banach space theory work. The detailed style of the book and the inclusion of background information make it easily accessible for beginning researchers, Ph.D. students, and non-specialists alike.

Introduction
1. Completely bounded and completely positive maps: basics
2. Completely bounded and completely positive maps: a tool kit
3. C*-algebras of discrete groups
4. C*-tensor products
5. Multiplicative domains of c.p. maps
6. Decomposable maps
7. Tensorizing maps and functorial properties
8. Biduals, injective von Neumann algebras and C*-norms
9. Nuclear pairs, WEP, LLP and QWEP
10. Exactness and nuclearity
11. Traces and ultraproducts
12. The Connes embedding problem
13. Kirchberg's conjecture
14. Equivalence of the two main questions
15. Equivalence with finite representability conjecture
16. Equivalence with Tsirelson's problem
17. Property (T) and residually finite groups. Thom's example
18. The WEP does not imply the LLP
19. Other proofs that C(n) < n. Quantum expanders
20. Local embeddability into ${mathscr{C}}$ and non-separability of $(OS_n, d_{cb})$
21. WEP as an extension property
22. Complex interpolation and maximal tensor product
23. Haagerup's characterizations of the WEP
24. Full crossed products and failure of WEP for $mathscr{B}otimes_{min}mathscr{B}$
25. Open problems
Appendix. Miscellaneous background
References
Index.

Subject Areas: Probability & statistics [PBT], Calculus & mathematical analysis [PBK]