Multipliers of commutative Banach algebras, power boundedness and Fourier-Stieltjes algebras

Abstract

Let A be a semisimple and regular commutative Banach algebra with bounded approximate identity. We study multipliers of A, in particular power bounded ones, and the associated ideals of A and A-invariant projections of the dual space of A. Samples of the results are general versions of the classical theorems of Choquet-Deny and of Foguel about measures on locally compact abelian groups. The results are linked to sets of synthesis in the Gelfand spectrum of A, and the main applications are concerned with Fourier and Fourier-Stieltjes algebras of locally compact groups.