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

Abstract

Let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ be a bounded operator between two noncommutative $L^p$-spaces, $1\leq p<\infty$. 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, $1\leq p\not=2 <\infty$, applies to an isometry $T\colon L^2({\mathcal M})\to L^2({\mathcal N})$ if and only if $T$ is $\ell^1$-contractive. We also show that a contractive operator $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ 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.