Separating Fourier and Schur multipliers

2023/03/24 by Arhancet, Cédric, Kriegler, Christoph, Merdy, Christian Le +1
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2303.13983

Abstract

Let G be a locally compact unimodular group, let 1≤ p<∞,let ϕ∈ L^∞(G) and assume that the Fourier multiplier M_ϕassociated with ϕ is bounded on the noncommutative L^p-space L^p(VN(G)).Then M_ϕ\colon 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 thereexists c∈\mathbb C and a continuouscharacter ψ\colon G→\mathbb C such that ϕ=cψ locally almost everywhere. This provides a characterization of isometricFourier multipliers on L^p(VN(G)), when p\not=2. Next, let Ω be a σ-finite measure space, let ϕ∈ L^∞(Ω²)and assume that the Schur multiplier associated with ϕ is bounded on the Schatten space S^p(L²(Ω)). We prove that this multiplier is separating if and only if there exist a constant c∈\mathbb C and two unitaries α,β∈ L^∞(Ω) such that ϕ(s,t) =c α(s)β(t) a.e. on Ω². This provides acharacterization of isometric Schur multiplierson S^p(L²(Ω)), when p\not=2.

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