Gilles Pisier is Emeritus Professor at Sorbonne Université and Distinguished Professor at Texas A & M University. He is the author of several books, including Introduction to Operator Space Theory (Cambridge, 2003) and Martingales in Banach Spaces (Cambridge, 2016). His multiple awards include the Salem prize in 1979 and the Ostrowski Prize in 1997, and he was the plenary speaker at the International Congress of Mathematicians in 1998.
Inhaltsangabe
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.
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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