Beurling-Fourier algebras on compact groups: spectral theory

TL;DR

This paper investigates the spectral properties of Beurling-Fourier algebras on compact groups, analyzing their spectra, symmetry, regularity, and spectral synthesis, extending classical Fourier algebra results with weighted variants.

Contribution

It introduces and studies the spectral theory of Beurling-Fourier algebras on compact groups, including spectrum characterization and conditions for symmetry and regularity.

Findings

Spectrum often equals the group for polynomial weights

Conditions for algebra symmetry and regularity are identified

Spectral synthesis results are established for various weights

Abstract

For a compact group $G$ we define the Beurling-Fourier algebra $A_\omega(G)$ on $G$ for weights $\omega$ defined on the dual $\what G$ and taking positive values. The classical Fourier algebra corresponds to the case $\omega$ is the constant weight 1. We study the Gelfand spectrum of the algebra realizing it as a subset of the complexification $G_{\mathbb C}$ defined by McKennon and Cartwright and McMullen. In many cases, such as for polynomial weights, the spectrum is simply $G$. We discuss the questions when the algebra $A_\omega(G)$ is symmetric and regular. We also obtain various results concerning spectral synthesis for $A_\omega(G)$.