Beurling–Fourier algebras on compact groups: Spectral theory

Abstract

For a compact group G we define the Beurling–Fourier algebra A ω ( G ) on G for weights ω : G ˆ → R > 0 . The classical Fourier algebra corresponds to the case ω is the constant weight 1. We study the Gelfand spectrum of the algebra realising it as a subset of the complexification G 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 ω ( G ) is symmetric and regular. We also obtain various results concerning spectral synthesis for A ω ( G ) .