Tensor products of C-star-algebras and operator spaces

Introduction page......Page 12 1.1 Completely bounded maps on operator spaces......Page 22 1.2 Extension property of B(H)......Page 29 1.3 Completely positive maps......Page 34 1.4 Normal c.p. maps on von Neumann algebras......Page 41 1.5 Injective operator algebras......Page 42 1.6 Factorization of completely bounded (c.b.) maps......Page 44 1.7 Normal c.b. maps on von Neumann algebras......Page 48 1.8 Notes and remarks......Page 50 2.1 Rows and columns: operator Cauchy–Schwarz inequality......Page 52 2.2 Automatic complete boundedness......Page 54 2.3 Complex conjugation......Page 55 2.4 Operator space dual......Page 59 2.5 Bi-infinite matrices with operator entries......Page 61 2.6 Free products of C*-algebras......Page 64 2.7 Universal C*-algebra of an operator space......Page 68 2.8 Completely positive perturbations of completely bounded maps......Page 69 2.9 Notes and remarks......Page 72 3.1 Full (=Maximal) group C*-algebras......Page 74 3.2 Full C*-algebras for free groups......Page 77 3.3 Reduced group C*-algebras: Fell’s absorption principle......Page 82 3.4 Multipliers......Page 84 3.5 Group von Neumann Algebra......Page 88 3.6 Amenable groups......Page 89 3.7 Operator space spanned by the free generators in C*λ(Fn)......Page 94 3.8 Free products of groups......Page 95 3.9 Notes and remarks......Page 96 4.1 C*-norms on tensor products......Page 98 4.2 Nuclear C*-algebras (a brief preliminary introduction)......Page 102 4.3 Tensor products of group C*-algebras......Page 103 4.4 A brief repertoire of examples from group C*-algebras......Page 106 4.5 States on the maximal tensor product......Page 107 4.6 States on the minimal tensor product......Page 110 4.7 Tensor product with a quotient C*-algebra......Page 114 4.8 Notes and remarks......Page 115 5.1 Multiplicative domains......Page 117 5.2 Jordan multiplicative domains......Page 119 5.3 Notes and remarks......Page 123 6.1 The dec-norm......Page 124 6.2 The δ-norm......Page 132 6.3 Decomposable extension property......Page 136 6.4 Examples of decomposable maps......Page 140 6.5 Notes and remarks......Page 146 7.1 (α → β)-tensorizing linear maps......Page 147 7.2 || ||max is projective (i.e. exact) but not injective......Page 152 7.3 max-injective inclusions......Page 155 7.4 || ||min is injective but not projective (i.e. not exact)......Page 161 7.5 min-projective surjections......Page 164 7.6 Generating new C*-norms from old ones......Page 168 7.7 Notes and remarks......Page 171 8.1 Biduals of C*-algebras......Page 172 8.2 The nor-norm and the bin-norm......Page 173 8.3 Nuclearity and injective von Neumann algebras......Page 174 8.4 Local reflexivity of the maximal tensor product......Page 181 8.5 Local reflexivity......Page 185 8.6 Notes and remarks......Page 190 9 Nuclear pairs, WEP, LLP, QWEP......Page 191 9.1 The fundamental nuclear pair (C*(F∞),B(l2))......Page 192 9.2 C*(F) is residually finite dimensional......Page 197 9.3 WEP (Weak Expectation Property)......Page 199 9.4 LLP (Local Lifting Property)......Page 204 9.5 To lift or not to lift (global lifting)......Page 209 9.6 Linear maps with WEP or LLP......Page 213 9.7 QWEP......Page 215 9.8 Notes and remarks......Page 219 10.1 The importance of being exact......Page 221 10.2 Nuclearity, exactness, approximation properties......Page 227 10.3 More on nuclearity and approximation properties......Page 233 10.4 Notes and remarks......Page 235 11.1 Traces......Page 236 11.2 Tracial probability spaces and the space L1(τ)......Page 239 11.3 The space L2(τ)......Page 241 11.4 An example from free probability: semicircular and circular systems......Page 246 11.5 Ultraproducts......Page 249 11.6 Factorization through B(H) and ultraproducts......Page 257 11.7 Hypertraces and injectivity......Page 267 11.8 The factorization property for discrete groups......Page 270 11.9 Notes and remarks......Page 272 12.1 Connes’s question......Page 273 12.2 The approximately finite dimensional (i.e. “hyperfinite”) II1-factor......Page 280 12.3 Hyperlinear groups......Page 282 12.4 Residually finite groups and Sofic groups......Page 284 12.5 Random matrix models......Page 287 12.6 Characterization of nuclear von Neumann algebras......Page 288 12.7 Notes and remarks......Page 290 13.1 LLP ⇒ WEP?......Page 291 13.2 Connection with Grothendieck’s theorem......Page 294 13.3 Notes and remarks......Page 301 14.1 From Connes’s question to Kirchberg’s conjecture......Page 302 14.2 From Kirchberg’s conjecture to Connes’s question......Page 303 14.3 Notes and remarks......Page 307 15.1 Finite representability conjecture......Page 308 15.2 Notes and remarks......Page 310 16.1 Unitary correlation matrices......Page 311 16.2 Correlation matrices with projection valued measures......Page 314 16.3 Strong Kirchberg conjecture......Page 320 16.4 Notes and remarks......Page 321 17 Property (T) and residually finite groups: Thom’s example......Page 322 17.1 Notes and remarks......Page 327 18 The WEP does not imply the LLP......Page 328 18.1 The constant C(n): WEP ⇒ LLP......Page 330 18.2 Proof that C(n) = √ n − 1 using random unitary matrices......Page 334 18.3 Exactness is not preserved by extensions......Page 338 18.4 A continuum of C*-norms on B⊗ B......Page 340 18.5 Notes and remarks......Page 343 19.1 Quantum coding sequences. Expanders. Spectral gap......Page 344 19.2 Quantum expanders......Page 347 19.3 Property (T)......Page 349 19.4 Quantum spherical codes......Page 352 19.5 Notes and remarks......Page 354 20 Local embeddability into C and nonseparability of (OSn,dcb)......Page 355 20.1 Perturbations of operator spaces......Page 356 20.2 Finite-dimensional subspaces of C......Page 357 20.3 Nonseparability of the metric space OSn of n-dimensional operator spaces......Page 362 20.4 Notes and remarks......Page 368 21.1 WEP as a local extension property......Page 369 21.2 WEP versus approximate injectivity......Page 373 21.3 The (global) lifting property LP......Page 375 21.4 Notes and remarks......Page 376 22.1 Complex interpolation......Page 377 22.2 Complex interpolation, WEP and maximal tensor product......Page 382 22.3 Notes and remarks......Page 393 23.1 Reduction to the σ-finite case......Page 395 23.2 A new characterization of generalized weak expectations and the WEP......Page 396 23.3 A second characterization of the WEP and its consequences......Page 399 23.4 Preliminaries on self-polar forms......Page 401 23.5 max+-injective inclusions and the WEP......Page 406 23.6 Complement......Page 414 23.7 Notes and remarks......Page 419 24.1 Full crossed products......Page 421 24.2 Full crossed products with inner actions......Page 425 24.3 B ⊗min B fails WEP......Page 429 24.4 Proof that C0(3) < 3 (Selberg’s spectral bound)......Page 438 24.5 Other proofs that C0(n) < n......Page 440 24.6 Random permutations......Page 442 24.7 Notes and remarks......Page 443 25 Open problems......Page 445 A.1 Banach space tensor products......Page 449 A.2 A criterion for an extension property......Page 450 A.4 Ultrafilters......Page 452 A.6 Finite representability......Page 454 A.7 Weak and weak* topologies: biduals of Banach spaces......Page 455 A.8 The local reflexivity principle......Page 457 A.9 A variant of Hahn–Banach theorem......Page 458 A.11 C*-algebras: basic facts......Page 459 A.12 Commutative C*-algebras......Page 461 A.13 States and the GNS construction......Page 462 A.14 On *-homomorphisms......Page 463 A.15 Approximate units, ideals, and quotient C*-algebras......Page 465 A.16 von Neumann algebras and their preduals......Page 467 A.17 Bitransposition: biduals of C*-algebras......Page 472 A.18 Isomorphisms between von Neumann algebras......Page 476 A.20 On σ-finite (countably decomposable) von Neumann algebras......Page 477 A.21 Schur’s lemma......Page 478 References......Page 481 Index......Page 493 "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 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 This advanced volume of lecture notes presents an open problem at the frontier of research into operator algebra theory, based on the author's university lecture courses and written in a widely accessible style for researchers and Ph.D. students with little experience in the area.