Separating Fourier and Schur Multipliers

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