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Complex Interpolation between Hilbert, Banach and Operator Spaces
Gilles Pisier, Texas A&M University, College Station, TX, and Université Paris VI, France

Memoirs of the American Mathematical Society
2010; 78 pp; softcover
Volume: 208
ISBN-10: 0-8218-4842-9
ISBN-13: 978-0-8218-4842-5
List Price: US$68
Individual Members: US$40.80
Institutional Members: US$54.40
Order Code: MEMO/208/978
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Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces \(X\) satisfying the following property: there is a function \(\varepsilon\to \Delta_X(\varepsilon)\) tending to zero with \(\varepsilon>0\) such that every operator \(T\colon \ L_2\to L_2\) with \(\|T\|\le \varepsilon\) that is simultaneously contractive (i.e., of norm \(\le 1\)) on \(L_1\) and on \(L_\infty\) must be of norm \(\le \Delta_X(\varepsilon)\) on \(L_2(X)\). The author shows that \(\Delta_X(\varepsilon) \in O(\varepsilon^\alpha)\) for some \(\alpha>0\) iff \(X\) is isomorphic to a quotient of a subspace of an ultraproduct of \(\theta\)-Hilbertian spaces for some \(\theta>0\) (see Corollary 6.7), where \(\theta\)-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).

Table of Contents

  • Introduction
  • Preliminaries. Regular operators
  • Regular and fully contractive operators
  • Remarks on expanding graphs
  • A duality operators/classes of Banach spaces
  • Complex interpolation of families of Banach spaces
  • \(\pmb{\theta}\)-Hilbertian spaces
  • Arcwise versus not arcwise
  • Fourier and Schur multipliers
  • A characterization of uniformly curved spaces
  • Extension property of regular operators
  • Generalizations
  • Operator space case
  • Generalizations (Operator space case)
  • Examples with the Haagerup tensor product
  • References
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