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A+ CATEGORY SCIENTIFIC UNIT

Real Interpolation between Row and Column Spaces

Volume 59 / 2011

Gilles Pisier Bulletin Polish Acad. Sci. Math. 59 (2011), 237-259 MSC: 46L07, 47L25, 46B70. DOI: 10.4064/ba59-3-6

Abstract

We give an equivalent expression for the -functional associated to the pair of operator spaces (R,C) formed by the rows and columns respectively. This yields a description of the real interpolation spaces for the pair (M_n(R), M_n(C)) (uniformly over n). More generally, the same result is valid when M_n (or B(\ell _2)) is replaced by any semi-finite von Neumann algebra. We prove a version of the non-commutative Khintchine inequalities (originally due to Lust-Piquard) that is valid for the Lorentz spaces L_{p,q}(\tau ) associated to a non-commutative measure \tau , simultaneously for the whole range 1\le p,q< \infty , regardless of whether p<2 or p>2. Actually, the main novelty is the case p=2, q\not =2. We also prove a certain simultaneous decomposition property for the operator norm and the Hilbert–Schmidt norm.

Authors

  • Gilles PisierMathematics Department
    Texas A&M University
    College Station, TX 77843, U.S.A.
    and
    Université Paris VI
    Institut Mathématique de Jussieu
    Analyse Fonctionnelle, Case 186
    75252 Paris Cedex 05, France
    e-mail

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