Let G be a locally compact unimodular group, let 1≤p<∞\documentclass[12pt]{minimal}
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\begin{document}$$1\le p<\infty $$\end{document}, let ϕ∈L∞(G)\documentclass[12pt]{minimal}
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\begin{document}$$\phi \in L^\infty (G)$$\end{document} and assume that the Fourier multiplier Mϕ\documentclass[12pt]{minimal}
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\begin{document}$$M_\phi $$\end{document} associated with ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} is bounded on the noncommutative Lp\documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}-space Lp(VN(G))\documentclass[12pt]{minimal}
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\begin{document}$$L^p(VN(G))$$\end{document}. Then MϕLp(VN(G))→Lp(VN(G))\documentclass[12pt]{minimal}
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\begin{document}$$M_\phi L^p(VN(G))\rightarrow L^p(VN(G))$$\end{document} is separating (that is, {a∗b=ab∗=0}⇒{Mϕ(a)∗Mϕ(b)=Mϕ(a)Mϕ(b)∗=0}\documentclass[12pt]{minimal}
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\begin{document}$$\{a^*b=ab^*=0\}\Rightarrow \{M_\phi (a)^* M_\phi (b)=M_\phi (a)M_\phi (b)^*=0\}$$\end{document} for any a,b∈Lp(VN(G))\documentclass[12pt]{minimal}
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\begin{document}$$a,b\in L^p(VN(G))$$\end{document}) if and only if there exists c∈C\documentclass[12pt]{minimal}
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\begin{document}$$c\in {\mathbb {C}}$$\end{document} and a continuous character ψG→C\documentclass[12pt]{minimal}
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\begin{document}$$\psi G\rightarrow {\mathbb {C}}$$\end{document} such that ϕ=cψ\documentclass[12pt]{minimal}
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\begin{document}$$\phi =c\psi $$\end{document} locally almost everywhere. This provides a characterization of isometric Fourier multipliers on Lp(VN(G))\documentclass[12pt]{minimal}
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\begin{document}$$L^p(VN(G))$$\end{document}, when p≠2\documentclass[12pt]{minimal}
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\begin{document}$$p\not =2$$\end{document}. Next, let Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} be a σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-finite measure space, let ϕ∈L∞(Ω2)\documentclass[12pt]{minimal}
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\begin{document}$$\phi \in L^\infty (\Omega ^2)$$\end{document} and assume that the Schur multiplier associated with ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} is bounded on the Schatten space Sp(L2(Ω))\documentclass[12pt]{minimal}
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\begin{document}$$S^p(L^2(\Omega ))$$\end{document}. We prove that this multiplier is separating if and only if there exist a constant c∈C\documentclass[12pt]{minimal}
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\begin{document}$$c\in {\mathbb {C}}$$\end{document} and two unitaries α,β∈L∞(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha ,\beta \in L^\infty (\Omega )$$\end{document} such that ϕ(s,t)=cα(s)β(t)\documentclass[12pt]{minimal}
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\begin{document}$$\phi (s,t) =c\, \alpha (s)\beta (t)$$\end{document} a.e. on Ω2.\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^2.$$\end{document} This provides a characterization of isometric Schur multipliers on Sp(L2(Ω))\documentclass[12pt]{minimal}
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\begin{document}$$S^p(L^2(\Omega ))$$\end{document}, when p≠2\documentclass[12pt]{minimal}
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\begin{document}$$p\not =2$$\end{document}.
