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

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.

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.