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Tensor Products of C*-Algebras and Operator Spaces
The Connes–Kirchberg Problem

Part of London Mathematical Society Student Texts

  • Date Published: February 2020
  • availability: Available
  • format: Paperback
  • isbn: 9781108749114


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About the Authors
  • 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 The reader is guided through a number of results (some of them previously unpublished) revolving around tensor products of C*-algebras and operator spaces, which are reminiscent of Grothendieck's famous Banach space theory work.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.

    • Contains new results previously known only to a few specialists
    • Makes all aspects of a major open problem accessible to beginning researchers, highlighting the connections with other subjects such as quantum information theory
    • Gives many examples, emphasizing the multiplicity of aspects and forms of the same problem
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    Product details

    • Date Published: February 2020
    • format: Paperback
    • isbn: 9781108749114
    • length: 494 pages
    • dimensions: 227 x 153 x 27 mm
    • weight: 0.7kg
    • availability: Available
  • Table of Contents

    1. Completely bounded and completely positive maps: basics
    2. Completely bounded and completely positive maps: a tool kit
    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 and 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. Thom's example
    18. The WEP does not imply the LLP
    19. Other proofs that C(n) < n. Quantum expanders
    20. Local embeddability into ${\mathscr{C}}$ and non-separability of $(OS_n, d_{cb})$
    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 $\mathscr{B}\otimes_{\min}\mathscr{B}$
    25. Open problems
    Appendix. Miscellaneous background

  • Author

    Gilles Pisier, Texas A & M University
    Gilles Pisier is Emeritus Professor at Sorbonne Université and Distinguished Professor at Texas A & M University. He is the author of several books, including Introduction to Operator Space Theory (Cambridge, 2003) and Martingales in Banach Spaces (Cambridge, 2016). His multiple awards include the Salem prize in 1979 and the Ostrowski Prize in 1997, and he was the plenary speaker at the International Congress of Mathematicians in 1998.

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