Ideals of the Fourier algebra, supports and harmonic operators

TL;DR

This paper investigates the null spaces of Herz-Schur multipliers and explores the structure of harmonic operators and functionals, providing new proofs and generalizations of existing results in harmonic analysis and operator theory.

Contribution

It introduces a unified approach to understanding supports of operators and extends previous results on harmonic operators using annihilation formulas.

Findings

Short proof and generalization of Neufang and Runde's result

Comparison of two notions of operator support

Characterization of jointly harmonic operators and functionals

Abstract

We examine the common null spaces of families of Herz-Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in J. Funct. Anal. 266 (2014), 6473-6500 can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.