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

ISBN-13: 9781108749114 / Angielski / Miękka / 2020 / 494 str.

Gilles Pisier

Opis

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

Recenzje

'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

Spis treści

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 ${\mathscr}$ and non-separability of $(OS_n, d_)$; 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 $\mathscr\otimes_{\min}\mathscr$; 25. Open problems; Appendix. Miscellaneous background; References; Index.