Separating Fourier and Schur Multipliers

Let G be a locally compact unimodular group, let \(1\le p<\infty \) $1\le p<\infty$ , let \(\phi \in L^\infty (G)\) $\varphi \in L^{\infty }(G)$ and assume that the Fourier multiplier \(M_\phi \) $M_{\varphi }$ associated with \(\phi \) $\varphi$ is bounded on the noncommutative \(L^p\) $L^p$ -space \(L^p(VN(G))\) $L^p(VN(G))$ . Then \(M_\phi L^p(VN(G))\rightarrow L^p(VN(G))\) $M_{\varphi }{L}^p(VN(G))\to L^p(VN(G))$ is separating (that is, \(\{a^*b=ab^*=0\}\Rightarrow \{M_\phi (a)^* M_\phi (b)=M_\phi (a)M_\phi (b)^*=0\}\) $\{a^{\ast }b=a{b}^{\ast }=0\}\Rightarrow \{M_{\varphi }{(a)}^{\ast }{M}_{\varphi }(b)=M_{\varphi }(a)M_{\varphi }{(b)}^{\ast }=0\}$ for any \(a,b\in L^p(VN(G))\) $a,b\in L^p(VN(G))$ ) if and only if there exists \(c\in {\mathbb {C}}\) $c\in \mathbb{C}$ and a continuous character \(\psi G\rightarrow {\mathbb {C}}\) $\psi G\to \mathbb{C}$ such that \(\phi =c\psi \) $\varphi =c\psi$ locally almost everywhere. This provides a characterization of isometric Fourier multipliers on \(L^p(VN(G))\) $L^p(VN(G))$ , when \(p\not =2\) $p\ne 2$ . Next, let \(\Omega \) $\mathrm{\Omega }$ be a \(\sigma \) $\sigma$ -finite measure space, let \(\phi \in L^\infty (\Omega ^2)\) $\varphi \in L^{\infty }({\mathrm{\Omega }}^2)$ and assume that the Schur multiplier associated with \(\phi \) $\varphi$ is bounded on the Schatten space \(S^p(L^2(\Omega ))\) $S^p(L^2(\mathrm{\Omega }))$ . We prove that this multiplier is separating if and only if there exist a constant \(c\in {\mathbb {C}}\) $c\in \mathbb{C}$ and two unitaries \(\alpha ,\beta \in L^\infty (\Omega )\) $\alpha ,\beta \in L^{\infty }(\mathrm{\Omega })$ such that \(\phi (s,t) =c\, \alpha (s)\beta (t)\) $\varphi (s,t)=c\alpha (s)\beta (t)$ a.e. on \(\Omega ^2.\) ${\mathrm{\Omega }}^2.$ This provides a characterization of isometric Schur multipliers on \(S^p(L^2(\Omega ))\) $S^p(L^2(\mathrm{\Omega }))$ , when \(p\not =2\) $p\ne 2$ .