On the spectra of multipliers

Abstract

In this note, some results concerning the spectral theory and Banach algebra properties of multipliers on compact Abelian groups are obtained. The study concentrates on multipliers whose spectra are, in a sense, natural, and whose transforms vanish at oo. The results are shown to be of particular interest in the case of the measure algebra M(G). Moreover, a necessary and sufficient condition is found for the spectrum of a Riesz product to equal the closure of the range of its Fourier-Stieltjes transform.