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

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 The reader is guided through a number of results (some of them previously unpublished) revolving around tensor products of C\*-algebras and operator spaces, which are reminiscent of Grothendieck's famous Banach space theory work.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. 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 "These notes are centered around the equivalence of two major open problems: one formulated by Connes (1976), about traces and ultraproducts of von Neumann algebras, the other one by Kirchberg (1993) about tensor products of C* algebras. This leads us to emphasize the notion of nuclear pair, that is a pair of C*-algebras admitting a unique tensor product. The main example is the pair (B,C) formed of the algebra B of bounded operators on Hilbert space and the full group C*-algebra C of any free group. This leads naturally to the weak expectation property (WEP) and the local lifting property (LLP), which we extensively study in connection with the more classical notions of nuclearity and exactness, or local reflexivity for C* algebras. We include two new characterizations of the WEP due to Haagerup but unpublished. We show that B fails the LLP and that the minimal tensor product of B with itself fails the WEP. Several properties of random unitary matrices and random permutations play a crucial role. We show the equivalence of the two main questions with a famous open one about Banach spaces and with Tsirelson's well known open problem in quantum information theory"-- Provided by publisher "These notes are centered around the equivalence of two major open problems: one formulated by Connes (1976), about traces and ultraproducts of von Neumann algebras, the other one by Kirchberg (1993) about tensor products of C* algebras. This leads us to emphasize the notion of nuclear pair, that is a pair of C*-algebras admitting a unique tensor product. The main example is the pair (B,C) formed of the algebra B of bounded operators on Hilbert space and the full group C*-algebra C of any free group. This leads naturally to the weak expectation property (WEP) and the local lifting property (LLP), which we extensively study in connection with the more classical notions of nuclearity and exactness, or local reflexivity for C* algebras. We include two new characterizations of the WEP due to Haagerup but unpublished. We show that B fails the LLP and that the minimal tensor product of B with itself fails the WEP. Several properties of random unitary matrices and random permutations play a crucial role. We show the equivalence of the two main questions with a famous open one about Banach spaces and with Tsirelson's well known open problem in quantum information theory"-- Résumé de l'éditeur