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Similarity Problems and Completely Bounded Maps

  • Book
  • © 1996

Overview

Authors:
  1. Gilles Pisier

    1. Mathematics Department, Texas A&M University, College Station, USA
      Equipe d’Analyse, Université Paris VI, Paris Cedex 05, France

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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1618)

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Table of contents (9 chapters)

  1. Front Matter

    Pages N2-vii
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  2. Introduction. Description of contents

    • Gilles Pisier
    Pages 1-12

  3. Von Neumann’s inequality and Ando’s generalization

    • Gilles Pisier
    Pages 13-29

  4. Non-unitarizable uniformly bounded group representations

    • Gilles Pisier
    Pages 30-52

  5. Completely bounded maps

    • Gilles Pisier
    Pages 53-69

  6. Completely bounded homomorphisms and derivations

    • Gilles Pisier
    Pages 70-91

  7. Schur multipliers and Grothendieck’s inequality

    • Gilles Pisier
    Pages 92-106

  8. Hankelian Schur multipliers. Herz-Schur multipliers

    • Gilles Pisier
    Pages 107-115

  9. The similarity problem for cyclic homomorphisms on a C*-algebra

    • Gilles Pisier
    Pages 116-132

  10. Completely bounded maps in the Banach space setting

    • Gilles Pisier
    Pages 133-142

  11. Back Matter

    Pages 143-161
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Keywords

About this book

These notes revolve around three similarity problems, appearing in three dif­ ferent contexts, but all dealing with the space B(H) of all bounded operators on a complex Hilbert space H. The first one deals with group representations, the second one with C* -algebras and the third one with the disc algebra. We describe them in detail in the introduction which follows. This volume is devoted to the background necessary to understand these three open problems, to the solutions that are known in some special cases and to numerous related concepts, results, counterexamples or extensions which their investigation has generated. For instance, we are naturally lead to study various Banach spaces formed by the matrix coefficients of group representations. Furthermore, we discuss the closely connected Schur multipliers and Grothendieck's striking characterization of those which act boundedly on B(H). While the three problems seem different, it is possible to place them in a common framework using the key concept of "complete boundedness", which we present in detail. In some sense, completely bounded maps can also be viewed as spaces of "coefficients" of C*-algebraic representations, if we allow "B(H)­ valued coefficients", this is the content of the fundamental factorization property of these maps, which plays a central role in this volume. Using this notion, the three problems can all be formulated as asking whether "boundedness" implies "complete boundedness" for linear maps satisfying cer­ tain additional algebraic identities.

Authors and Affiliations

  • Mathematics Department, Texas A&M University, College Station, USA

    Gilles Pisier

  • Equipe d’Analyse, Université Paris VI, Paris Cedex 05, France

    Gilles Pisier

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