Separating Fourier and Schur multipliers

Cédric Arhancet, Christoph Kriegler, Christian Le Merdy*, Safoura Zadeh

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

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(V N(G)). Then Mφ : Lp(V N(G)) → Lp(V N(G)) is separating (that is, {ab = ab = 0}⇒{Mφ(a)Mφ(b) = Mφ(a)Mφ(b) = 0} for any a, b ∈ Lp(V N(G))) if and only if there exists cC and a continuous character ψ : GC such that φ = locally almost everywhere. This provides a characterization of isometric Fourier multipliers on Lp(V N(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 cC and two unitaries α, β ∈ L(Ω) such that φ(s, t) = c α(s)β(t) a.e. on Ω2. This provides a characterization of isometric Schur multipliers on Sp(L2(Ω)), when p ≠
 2.
Original languageEnglish
Article number5
Number of pages27
JournalJournal of Fourier Analysis and Applications
Volume30
Issue number5
DOIs
Publication statusPublished - 2 Jan 2024

Bibliographical note

Funding Information:
Cédric Arhancet and Christoph Kriegler were supported by the grant ANR-18-CE40-0021 of the French National Research Agency ANR (project HASCON). Christoph Kriegler was supported by the grant ANR-17-CE40-0021 (project Front). Christian Le Merdy was supported by the ANR project Noncommutative analysis on groups and quantum groups (No./ANR-19-CE40-0002). Safoura Zadeh was supported by I-SITE Emergence project MultiStructure (Harmonic Analysis of Fourier and Schur multipliers) of Clermont Auvergne University.

Publisher Copyright:
© 2024, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

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