Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 pp. 417-442.

$\ell^1$ -contractive maps on noncommutative $L^p$ -spaces

Authors: Christian Le Merdy (1), Safoura Zadeh (2)
Author institution: (1) Laboratoire de Mathematiques de Besancon, Universite Bourgogne Franche-Comte, France
(2) Department of Mathematics, Federal University of Paraiba, Brazil

Summary: Let $T: L^p(\mathcal M)\to L^p(\mathcal N)$ be a bounded operator between two noncommutative $L^p$ -spaces, $1\leqslant p<\infty$ . We say that $T$ is $\ell^1$ -bounded (respectively $\ell^1$ -contractive) if $T\otimes I_{\ell^1}$ extends to a bounded (respectively contractive) map from $L^p(\mathcal M;\ell^1)$ into $L^p(\mathcal N;\ell^1)$ . We show that Yeadon's factorization theorem for $L^p$ -isometries, $1\leqslant p\not=2 <\infty$ , applies to an isometry $T: L^2(\mathcal M)\to L^2(\mathcal N)$ if and only if $T$ is $\ell^1$ -contractive. We also show that a contractive operator $T: L^p(\mathcal M)\to L^p(\mathcal N)$ is automatically $\ell^1$ -contractive if it satisfies one of the following two conditions: either $T$ is $2$ -positive; or $T$ is separating, that is, for any disjoint $a,b\in L^p(\mathcal M)$ (i.e.\ $a^*b=ab^*=0)$ , the images $T(a),T(b)$ are disjoint as well.

DOI: http://dx.doi.org/10.7900/jot.2019oct09.2257
Keywords: noncommutative $L^p$ -spaces, regular maps, positive maps, isometries

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