# 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