Real Interpolation between Row and Column Spaces

Gilles Pisier¹

¹ Mathematics 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

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) no. 3, pp. 237-259

Résumé

We give an equivalent expression for the $K$-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 $p2 $ 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.

Détail

DOI : 10.4064/ba59-3-6

Keywords: equivalent expression k functional associated pair operator spaces formed rows columns respectively yields description real interpolation spaces pair uniformly generally result valid ell replaced semi finite von nbsp neumann algebra prove version non commutative khintchine inequalities originally due lust piquard valid lorentz spaces tau associated non commutative measure tau simultaneously whole range infty regardless whether actually main novelty prove certain simultaneous decomposition property operator norm hilbert schmidt norm