Introduction to Operator Space Theory (Broché)

Gilles Pisier

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  • Cambridge University Press

  • Paru le : 01/12/2003
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The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An " operator space " is simply a Banach space with an embedding into the space B (H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate varions aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C*-algebras. The third (and shortest) part of the book describes applications to non-self-adjoint operator algebras and similarity problems. In particular, the author's counterexample to the " Halmos problem " is presented, as well as work on the new concept of "length" of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras, and theoretical physics will find that this book has much to offer.
  • INTRODUCTION TO OPERATOR SPACES
    • Completely bounded maps
    • Minimal and maximal operator space structures
    • Projective tensor product
    • The Haagerup tensor product
    • Characterizations of operator algebras
    • The operator Hilbert space
    • Group C*-algebra - Universal algebars and unitization for an operator space
    • Examples and comments
    • Comparisons
  • OPERATOR SPACES AND C*-TENSOR PRODUCTS
    • C*-norms on tensor products - Decomposable maps - Nuclearity
    • Nuclearity and approximation properties
  • C*
    • Kirchberg's theorem on decomposable maps
    • The Weak Expectation Property (WEP)
    • The Local Lifting Property (LLP)
    • Exactness
    • Local reflexivity
    • Grothendieck's theorem for operator spaces
    • Estimating the norms of sums of unitaries: Ramanujan graphs, property T, random matrices
    • Local theory of operator spaces
  • B (H)
    • Completely isomorphic C*-algebras
    • Injective and projective operator spaces
  • OPERATOR SPACES AND NON-SELF-ADJOINT: OPERATOR ALGEBRAS
    • Maximal tensor products and free products of operator algebras
    • The Blecher-Paulsen factorization - Infinite Haagerup tensor products
    • Similitary problems
    • The Sz - Nagy-Halmos similitary problems

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