Some ergodic properties of multipliers on commutative Banach algebras

A commutative semisimple regular Banach algebra $A$ with the Gelfand space $ \Sigma _{A}$ is called a Ditkin algebra if each point of $\Sigma _{A}\cup \left\{ \infty \right\} $ is a set of synthesis for $A$. Generalizing the Choquet-Deny theorem, it is shown that if $T$ is a multiplier of a Ditkin algebra $A,$ then $\left\{ \varphi \in A^{\ast }:T^{\ast }\varphi =\varphi \right\} $ is finite dimensional if and only if \textnormal{card}$\mathcal{F}_{T}$ is finite, where $\mathcal{F}_{T}=\left\{ \gamma \in \Sigma _{A}:\widehat{T}\left( \gamma \right) =1\right\} $ and $ \widehat{T}$ is the Helgason-Wang representation of $T.$