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

Abstract

Let $T:L^p(\mathcal{M})\to L^p(\mathcal{N})$ be a bounded operator between two noncommutative $L^p$-spaces, $1\le p<\infty$. We say that $T$ is ${\ell }^1$-bounded (resp. ${\ell }^1$-contractive) if $T\otimes I_{{\ell }^1}$ extends to a bounded (resp. 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, $1\le p\ne 2<\infty$, 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. $a^{\ast }b=a{b}^{\ast }=0)$, the images $T(a),T(b)$ are disjoint as well.

This is a revised version with a few corrections. To appear in Journal of Operator Theory

References in corpus (1)

• Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps

Cited by in corpus (1)

• A characterization of completely bounded normal Jordan $\ast$-homomorphisms on von Neumann algebras