Separating Fourier and Schur multipliers

Abstract

Let $G$ be a locally compact unimodular group, let $1\le p<\infty$,let $\varphi \in L^{\infty }(G)$ and assume that the Fourier multiplier $M_{\varphi }$associated with $\varphi$ is bounded on the noncommutative $L^p$-space $L^p(VN(G))$.Then $M_{\varphi }:L^p(VN(G))\to L^p(VN(G))$ is separating (that is,$\{a^{\ast }b=a{b}^{\ast }=0\}\Rightarrow \{M_{\varphi }(a{)}^{\ast }{M}_{\varphi }(b)=M_{\varphi }(a)M_{\varphi }(b{)}^{\ast }=0\}$for any $a,b\in L^p(VN(G))$) if and only if thereexists $c\in \mathbb{C}$ and a continuouscharacter $\psi :G\to \mathbb{C}$ such that $\varphi =c\psi$ locally almost everywhere. This provides a characterization of isometricFourier multipliers on $L^p(VN(G))$, when $p\ne 2$. Next, let $\Omega$ be a $\sigma$-finite measure space, let $\varphi \in L^{\infty }({\Omega }^2)$and assume that the Schur multiplier associated with $\varphi$ is bounded on the Schatten space $S^p(L^2(\Omega ))$. We prove that this multiplier is separating if and only if there exist a constant $c\in \mathbb{C}$ and two unitaries $\alpha ,\beta \in L^{\infty }(\Omega )$ such that $\varphi (s,t)=c\alpha (s)\beta (t)$ a.e. on ${\Omega }^2.$ This provides acharacterization of isometric Schur multiplierson $S^p(L^2(\Omega ))$, when $p\ne 2$.