Some ergodic properties of multipliers on commutative Banach algebras

Abstract

A commutative semisimple regular Banach algebra A with the Gelfand space $textstylesum_A$ is called a Ditkin algebraif each point of ${textstylesum_A}cup{infty}$ is a set of synthesis for A. Generalizing the Choquet–Deny theorem, it is shown that if T isa multiplier of a Ditkin algebra A; then ${rhoin A^ast:T^astrho=rho}$ is finite dimensional if and only if card $F_T$ is finite, where $F_T={gammain{textstylesum_A}:overline T(y)=1}$ and $overline T$ is the Helgason–Wang representation of T.