Book contents
- Frontmatter
- Contents
- Introduction
- 1 Completely bounded and completely positive maps
- 2 Completely bounded and completely positive maps
- 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
- 18 The WEP does not imply the LLP
- 19 Other proofs that C(n)
- 20
Local embeddability into C and nonseparability of (OSn, dcb)- 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 ⊗min B- 25
Open problems- Appendix
Miscellaneous backgroundReferencesIndex - 20
23 - Haagerup’s characterizations of the WEP
Published online by Cambridge University Press: 10 February 2020
Book contents
- Frontmatter
- Contents
- Introduction
- 1 Completely bounded and completely positive maps
- 2 Completely bounded and completely positive maps
- 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
- 18 The WEP does not imply the LLP
- 19 Other proofs that C(n)
- 20 Local embeddability into C and nonseparability of (OSn, dcb)
- 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 ⊗min B
- 25 Open problems
- Appendix Miscellaneous background
- References
- Index
Summary
This chapter starts with an overview of the complex interpolation method, for pairs of Banach spaces. Our main application here is when the pair is formed of the same space X with two equivalent norms. Fix an integer n. We consider a C*-algebra A and the space X formedof n-tuples in A equipped with two norms: the row-norm and the column-norm. In that case we prove a remarkable formula identifying the interpolated norm of parameter 1/2 (the midpoint of the interpolation scale). The latter formula involves the maximal tensor product of A with its complex conjugate. This is a preparation for the next chapter.
- Type
- Chapter
- Information
- Tensor Products of C*-Algebras and Operator SpacesThe Connes–Kirchberg Problem, pp. 384 - 409Publisher: Cambridge University PressPrint publication year: 2020