Idempotents of small norm

TL;DR

This paper characterizes idempotent functions of small norm in harmonic analysis, showing that such functions correspond to unions of cosets or open cosets in locally compact groups, depending on the group type.

Contribution

It proves new bounds on the norms of idempotents in Fourier and Fourier-Stieltjes algebras, resolving open questions for both abelian and non-abelian groups.

Findings

Idempotents with norm less than 4/3 in abelian groups are unions of two cosets.

Idempotents with cb-norm less than (1+√2)/2 in general groups are open cosets.

Provides a complete characterization of small-norm idempotents in harmonic analysis.

Abstract

Let $\Gamma$ be a locally compact group. We answer two questions left open in [7] and [9]: i) For abelian $\Gamma$, we prove that if $\chi_S \in B(\Gamma)$ is an idempotent with norm $\left\|\chi_S \right\| < \frac{4}{3}$, then $S$ is the union of two cosets of an open subgroup of $\Gamma$. ii) For general $\Gamma$, we prove that if $\chi_S \in M_{cb}A(\Gamma)$ is an idempotent with norm $\left\| \chi_S \right\|_{cb} < \frac{1 + \sqrt{2}}{2}$, then $S$ is an open coset in $\Gamma$.