Tensor Products of C*-Algebras and Operator Spaces: The Connes–Kirchberg Problem (London Mathematical Society Student Texts, Band 96)

Introduction page
1 Completely bounded and completely positive maps: Basics
	1.1 Completely bounded maps on operator spaces
	1.2 Extension property of B(H)
	1.3 Completely positive maps
	1.4 Normal c.p. maps on von Neumann algebras
	1.5 Injective operator algebras
	1.6 Factorization of completely bounded (c.b.) maps
	1.7 Normal c.b. maps on von Neumann algebras
	1.8 Notes and remarks
2 Completely bounded and completely positive maps: A tool kit
	2.1 Rows and columns: operator Cauchy–Schwarz inequality
	2.2 Automatic complete boundedness
	2.3 Complex conjugation
	2.4 Operator space dual
	2.5 Bi-infinite matrices with operator entries
	2.6 Free products of C*-algebras
	2.7 Universal C*-algebra of an operator space
	2.8 Completely positive perturbations of completely bounded maps
	2.9 Notes and remarks
3 C*-algebras of discrete groups
	3.1 Full (=Maximal) group C*-algebras
	3.2 Full C*-algebras for free groups
	3.3 Reduced group C*-algebras: Fell’s absorption principle
	3.4 Multipliers
	3.5 Group von Neumann Algebra
	3.6 Amenable groups
	3.7 Operator space spanned by the free generators in C*λ(Fn)
	3.8 Free products of groups
	3.9 Notes and remarks
4 C*-tensor products
	4.1 C*-norms on tensor products
	4.2 Nuclear C*-algebras (a brief preliminary introduction)
	4.3 Tensor products of group C*-algebras
	4.4 A brief repertoire of examples from group C*-algebras
	4.5 States on the maximal tensor product
	4.6 States on the minimal tensor product
	4.7 Tensor product with a quotient C*-algebra
	4.8 Notes and remarks
5 Multiplicative domains of c.p. maps
	5.1 Multiplicative domains
	5.2 Jordan multiplicative domains
	5.3 Notes and remarks
6 Decomposable maps
	6.1 The dec-norm
	6.2 The δ-norm
	6.3 Decomposable extension property
	6.4 Examples of decomposable maps
	6.5 Notes and remarks
7 Tensorizing maps and functorial properties
	7.1 (α → β)-tensorizing linear maps
	7.2 ||  ||max is projective (i.e. exact) but not injective
	7.3 max-injective inclusions
	7.4 ||  ||min is injective but not projective (i.e. not exact)
	7.5 min-projective surjections
	7.6 Generating new C*-norms from old ones
	7.7 Notes and remarks
8 Biduals, injective von Neumann algebras, and C*-norms
	8.1 Biduals of C*-algebras
	8.2 The nor-norm and the bin-norm
	8.3 Nuclearity and injective von Neumann algebras
	8.4 Local reflexivity of the maximal tensor product
	8.5 Local reflexivity
	8.6 Notes and remarks
9 Nuclear pairs, WEP, LLP, QWEP
	9.1 The fundamental nuclear pair (C*(F∞),B(ℓ2))
	9.2 C*(F) is residually finite dimensional
	9.3 WEP (Weak Expectation Property)
	9.4 LLP (Local Lifting Property)
	9.5 To lift or not to lift (global lifting)
	9.6 Linear maps with WEP or LLP
	9.7 QWEP
	9.8 Notes and remarks
10 Exactness and nuclearity
	10.1 The importance of being exact
	10.2 Nuclearity, exactness, approximation properties
	10.3 More on nuclearity and approximation properties
	10.4 Notes and remarks
11 Traces and ultraproducts
	11.1 Traces
	11.2 Tracial probability spaces and the space L1(τ)
	11.3 The space L2(τ)
	11.4 An example from free probability: semicircular and circular systems
	11.5 Ultraproducts
	11.6 Factorization through B(H) and ultraproducts
	11.7 Hypertraces and injectivity
	11.8 The factorization property for discrete groups
	11.9 Notes and remarks
12 The Connes embedding problem
	12.1 Connes’s question
	12.2 The approximately finite dimensional (i.e. “hyperfinite”) II1-factor
	12.3 Hyperlinear groups
	12.4 Residually finite groups and Sofic groups
	12.5 Random matrix models
	12.6 Characterization of nuclear von Neumann algebras
	12.7 Notes and remarks
13 Kirchberg’s conjecture
	13.1 LLP ⇒ WEP?
	13.2 Connection with Grothendieck’s theorem
	13.3 Notes and remarks
14 Equivalence of the two main questions
	14.1 From Connes’s question to Kirchberg’s conjecture
	14.2 From Kirchberg’s conjecture to Connes’s question
	14.3 Notes and remarks
15 Equivalence with finite representability conjecture
	15.1 Finite representability conjecture
	15.2 Notes and remarks
16 Equivalence with Tsirelson’s problem
	16.1 Unitary correlation matrices
	16.2 Correlation matrices with projection valued measures
	16.3 Strong Kirchberg conjecture
	16.4 Notes and remarks
17 Property (T) and residually finite groups: Thom’s example
	17.1 Notes and remarks
18 The WEP does not imply the LLP
	18.1 The constant C(n): WEP ⇒ LLP
	18.2 Proof that C(n) = √ n − 1 using random unitary matrices
	18.3 Exactness is not preserved by extensions
	18.4 A continuum of C*-norms on B⊗ B
	18.5 Notes and remarks
19 Other proofs that C(n)< n: quantum expanders
	19.1 Quantum coding sequences. Expanders. Spectral gap
	19.2 Quantum expanders
	19.3 Property (T)
	19.4 Quantum spherical codes
	19.5 Notes and remarks
20 Local embeddability into C and nonseparability of (OSn,dcb)
	20.1 Perturbations of operator spaces
	20.2 Finite-dimensional subspaces of C
	20.3 Nonseparability of the metric space OSn of n-dimensional operator spaces
	20.4 Notes and remarks
21 WEP as an extension property
	21.1 WEP as a local extension property
	21.2 WEP versus approximate injectivity
	21.3 The (global) lifting property LP
	21.4 Notes and remarks
22 Complex interpolation and maximal tensor product
	22.1 Complex interpolation
	22.2 Complex interpolation, WEP and maximal tensor product
	22.3 Notes and remarks
23 Haagerup’s characterizations of the WEP
	23.1 Reduction to the σ-finite case
	23.2 A new characterization of generalized weak expectations and the WEP
	23.3 A second characterization of the WEP and its consequences
	23.4 Preliminaries on self-polar forms
	23.5 max+-injective inclusions and the WEP
	23.6 Complement
	23.7 Notes and remarks
24 Full crossed products and failure of WEP for B ⊗min B
	24.1 Full crossed products
	24.2 Full crossed products with inner actions
	24.3 B ⊗min B fails WEP
	24.4 Proof that C0(3) < 3 (Selberg’s spectral bound)
	24.5 Other proofs that C0(n) < n
	24.6 Random permutations
	24.7 Notes and remarks
25 Open problems
Appendix: Miscellaneous background
	A.1 Banach space tensor products
	A.2 A criterion for an extension property
	A.3 Uniform convexity of Hilbert space
	A.4 Ultrafilters
	A.5 Ultraproducts of Banach spaces
	A.6 Finite representability
	A.7 Weak and weak* topologies: biduals of Banach spaces
	A.8 The local reflexivity principle
	A.9 A variant of Hahn–Banach theorem
	A.10 The trace class
	A.11 C*-algebras: basic facts
	A.12 Commutative C*-algebras
	A.13 States and the GNS construction
	A.14 On *-homomorphisms
	A.15 Approximate units, ideals, and quotient C*-algebras
	A.16 von Neumann algebras and their preduals
	A.17 Bitransposition: biduals of C*-algebras
	A.18 Isomorphisms between von Neumann algebras
	A.19 Tensor product of von Neumann algebras
	A.20 On σ-finite (countably decomposable) von Neumann algebras
	A.21 Schur’s lemma
References
Index