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Factorization of Linear Operators and Geometry of Banach Spaces
Gilles Pisier
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1986; 154 pp; softcover
Number: 60
Reprint/Revision History:
reprinted 1987
ISBN-10: 0-8218-0710-2
ISBN-13: 978-0-8218-0710-1
List Price: US$29
Member Price: US$23.20
All Individuals: US$23.20
Order Code: CBMS/60
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This book surveys the considerable progress made in Banach space theory as a result of Grothendieck's fundamental paper Reśumé de la théorie métrique des produits tensoriels topologiques. The author examines the central question of which Banach spaces \(X\) and \(Y\) have the property that every bounded operator from \(X\) to \(Y\) factors through a Hilbert space, in particular when the operators are defined on a Banach lattice, a \(C^*\)-algebra or the disc algebra and \(H^\infty\). He reviews the six problems posed at the end of Grothendieck's paper, which have now all been solved (except perhaps the exact value of Grothendieck's constant), and includes the various results which led to their solution. The last chapter contains the author's construction of several Banach spaces such that the injective and projective tensor products coincide; this gives a negative solution to Grothendieck's sixth problem.

Although the book is aimed at mathematicians working in functional analysis, harmonic analysis and operator algebras, its detailed and self-contained treatment makes the material accessible to nonspecialists with a grounding in basic functional analysis. In fact, the author is particularly concerned to develop very recent results in the geometry of Banach spaces in a form that emphasizes how they may be applied in other fields, such as harmonic analysis and \(C^*\)-algebras.

Table of Contents

  • Absolutely summing operators and basic applications
  • Factorization through a Hilbert space
  • Type and cotype. Kwapien's theorem
  • The "abstract" version of Grothendieck's theorem
  • Grothendieck's theorem
  • Banach spaces satisfying Grothendieck's theorem
  • Applications of the volume ratio method
  • Banach lattices
  • \(C^*\)-algebras
  • Counterexamples to Grothendieck's conjecture
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