Beurling–Fourier algebras on compact groups: Spectral theory

For a compact group G we define the Beurling–Fourier algebra Aω(G) on G for weights . 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 GC 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 Aω(G) is symmetric and regular. We also obtain various results concerning spectral synthesis for Aω(G).