Beurling-Fourier algebras on compact groups: Spectral theory

Journal article, 2012

For a compact group G we define the Beurling-Fourier algebra A(omega)(G) on G for weights omega: (G) over cap -> R(>0). The classical Fourier algebra corresponds to the case omega is the constant weight I. 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(omega)(G) is symmetric and regular. We also obtain various results concerning spectral synthesis for A(omega)(G).