Tensor products of C★-algebras and operator spaces : the Connes-Kirchberg problem /
Gilles Pisier.
Bok Engelsk 2020 · Electronic books.
| Annen tittel | |
|---|---|
| Utgitt | Cambridge University Press
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| Omfang | 1 online resource (x, 484 pages) : : digital, PDF file(s).
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| Opplysninger | Title from publisher's bibliographic system (viewed on 07 Feb 2020).. - Cover -- Series information -- Title page -- Copyright information -- Contents -- Introduction -- 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*[sub(λ)](F[sub(n)]) -- 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.. - 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 [nRightarrow] LLP -- 18.2 Proof that C(n) = 2[sqrt(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 (OS[sub(n)],d[sub(cb)]) -- 20.1 Perturbations of operator spaces -- 20.2 Finite-dimensional subspaces of C -- 20.3 Nonseparability of the metric space OS[sub(n)] 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[sup(+)]-injective inclusions and the WEP -- 23.6 Complement -- 23.7 Notes and remarks -- 24 Full crossed products and failure of WEP for B ⊗[sub(min)] B -- 24.1 Full crossed products -- 24.2 Full crossed products with inner actions -- 24.3 B ⊗[sub(min)] B fails WEP -- 24.4 Proof that C[sub(0)](3) < -- 3 (Selberg's spectral bound) -- 24.5 Other proofs that C[sub(0)](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.. - 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 || || [sub(max)] is projective (i.e. exact) but not injective -- 7.3 max-injective inclusions -- 7.4 || ||[sub(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(ell[sub(2)])) -- 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 L[sub(1)](τ) -- 11.3 The space L[sub(2)](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")II[sub(1)]-factor.. - 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.
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| Emner | |
| Sjanger | |
| Dewey | |
| ISBN | 1-108-78208-6
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