Separating Fourier and Schur Multipliers

Let G be a locally compact unimodular group, let 1≤ p<∞ , let ϕ∈ L^∞ (G) and assume that the Fourier multiplier M_ϕ associated with ϕ is bounded on the noncommutative L^p -space L^p(VN(G)) . Then M_ϕ L^p(VN(G))→ L^p(VN(G)) is separating (that is, {a^*b=ab^*=0}⇒{M_ϕ (a)^* M_ϕ (b)=M_ϕ (a)M_ϕ (b)^*=0} for any a,b∈ L^p(VN(G)) ) if and only if there exists c∈ℂ and a continuous character ψ G→ℂ such that ϕ =cψ locally almost everywhere. This provides a characterization of isometric Fourier multipliers on L^p(VN(G)) , when p≠2 . Next, let Ω be a σ -finite measure space, let ϕ∈ L^∞ (Ω ^2) and assume that the Schur multiplier associated with ϕ is bounded on the Schatten space S^p(L^2(Ω )) . We prove that this multiplier is separating if and only if there exist a constant c∈ℂ and two unitaries α ,β∈ L^∞ (Ω ) such that ϕ (s,t) =c α (s)β (t) a.e. on Ω ^2. This provides a characterization of isometric Schur multipliers on S^p(L^2(Ω )) , when p≠2 .