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  • © 1996

Similarity Problems and Completely Bounded Maps

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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
  2. Introduction. Description of contents

    • Gilles Pisier
    Pages 1-12
  3. Completely bounded maps

    • Gilles Pisier
    Pages 53-69
  4. Back Matter

    Pages 143-161

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.

Keywords

  • 42B30
  • 43A65
  • 46E40
  • 46L05
  • 46L57
  • 47A05
  • 47A20
  • 47B10
  • 47C15
  • 47L20
  • Completely bounded representation
  • H (p) spaces
  • Harmonic analysis
  • Polynomially bounded operator
  • operator algebras
  • similar to contraction
  • similar to unitary
  • uniformly bounded representation
  • operator algebra
  • solution

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

Bibliographic Information

Buying options

eBook USD 74.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions