Tensor products of C*-algebras and operator spaces : the Connes-Kirchberg problem
Author(s)
Bibliographic Information
Tensor products of C*-algebras and operator spaces : the Connes-Kirchberg problem
（London Mathematical Society student texts, 96）
Cambridge University Press, 2020
- : hbk
- : pbk
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
PIS||3||3200040100891
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Note
Includes bibliographical references (p. 470-481) and index
Description and Table of Contents
Description
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 The reader is guided through a number of results (some of them previously unpublished) revolving around tensor products of C*-algebras and operator spaces, which are reminiscent of Grothendieck's famous Banach space theory work.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.
Table of Contents
- 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.
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