Separating Fourier and Schur multipliers
Multiplicateurs de Fourier et de Schur séparants
Résumé
Let G be a locally compact unimodular group, let 1≤p<∞,
let ϕ∈L∞(G) and assume that the Fourier multiplier Mϕ
associated with ϕ is bounded on the noncommutative Lp-space Lp(VN(G)).
Then Mϕ:Lp(VN(G))→Lp(VN(G)) is separating (that is,
{a∗b=ab∗=0}⇒{Mϕ(a)∗Mϕ(b)=Mϕ(a)Mϕ(b)∗=0}
for any a,b∈Lp(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 Lp(VN(G)), when p≠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 Sp(L2(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)=cα(s)β(t) a.e. on
Ω2. This provides a
characterization of isometric Schur multipliers
on Sp(L2(Ω)), when p≠2.
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