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 ∞. 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.
