Tensor Products of C*-Algebras and Operator Spaces

The Connes–Kirchberg Problem

Gilles Pisier

• Publisher: Cambridge University Press
• Year: 2020
• Print ISBN:
  9781108479011
• eBook ISBN:
  9781108803205
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

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 The reader is guided through a number of results (some of them previously unpublished) revolving around tensor products of C*-algebras and operator spaces, which are reminiscent of Grothendieck's famous Banach space theory work.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

• Cover
• Series information
• Title page
• Copyright information
• 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, 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 C and nonseparability of (OS[sub(n)],d[sub(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 B ⊗[sub(min)] B
• 25 Open problems
• Appendix Miscellaneous background
• References
• Index