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
14 - Equivalence of the two main questions
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
One of Kirchberg’s conjecture that we emphasize here is whether the LLP implies the WEP. This actually reduces to the case of the full C* algebra C of the free group with countably infinitely many generators, which is the prototypical example with the LLP. The question is shown to be equivalent to a very simple inequality, involving the linear span of the unitary generators of C, that seems to be related to Grothendieck’s classicalinequality from Banach space theory. Various results are proved that tend to « almost prove » the conjecture, notably one by Tsirelson in which it would suffice to replace real scalars by complex ones to obtain the full conjecture.
- Type
- Chapter
- Information
- Tensor Products of C*-Algebras and Operator SpacesThe Connes–Kirchberg Problem, pp. 291 - 296Publisher: Cambridge University PressPrint publication year: 2020