## Abstract

Let G be a locally compact unimodular group, let 1 ≤

2. Next, let

2.

*p*< ∞, let*φ*∈*L*and assume that the Fourier multiplier^{∞}(G)*M*associated with_{φ}*φ*is bounded on the noncommutative*L*-space^{p}*L*. Then^{p}(V N(G))*M*:_{φ}*L*→^{p}(V N(G))*L*is separating (that is, {^{p}(V N(G))*a*=^{∗}b*ab*= 0}⇒{^{∗}*M*(_{φ}*a*)^{∗}*M*(_{φ}*b*) =*M*(_{φ}*a*)*M*(_{φ}*b*)^{∗}= 0} for any*a, b*∈*L*) if and only if there exists^{p}(V N(G))*c*∈**and a continuous character***C**ψ*:*G*→**such that***C**φ*=*cψ*locally almost everywhere. This provides a characterization of isometric Fourier multipliers on*L*, when p ≠^{p}(V N(G))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*∈**and two unitaries***C**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.

Original language | English |
---|---|

Article number | 5 |

Number of pages | 27 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 30 |

Issue number | 5 |

DOIs | |

Publication status | Published - 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.