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
1 - Completely bounded and completely positive maps
Published online by Cambridge University Press: 10 February 2020
- 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
In the remarkable paper where he proved the equivalence, Kirchberg studied more generally the pairs of C*-algebras(A,B) admitting only one C*-norm on their algebraic tensor product.We call such pairs "nuclear pairs''. A C*-algebra A istraditionally called nuclear if this holds for any C*-algebra B. Our exposition chooses as its cornerstone Kirchberg's theoremasserting the nuclearity of what is for us the "fundamental pair'', namely the pair (B,C)where B is the algebra B of bounded operators on Hilbert space and C isthe full group C*-algebra C of the free group with countably infinitely many generators. Our presentation leads us to highlight two properties of C*-algebras, the Weak Expectation Property (WEP) and the Local Lifting Property (LLP).
- Type
- Chapter
- Information
- Tensor Products of C*-Algebras and Operator SpacesThe Connes–Kirchberg Problem, pp. 11 - 40Publisher: Cambridge University PressPrint publication year: 2020