$A$-Ergodicity of Convolution Operators in Group Algebras

Let $G$ be a locally compact Abelian group with dual group $\Gamma $, let $\mu$ be a power bounded measure on $G$, and let $A=[ a_{n,k}]_{n,k=0}^{\infty}$ be a strongly regular matrix. We show that the sequence $\{\sum_{k=0}^{\infty}a_{n,k}\mu^{k}\ast f\}_{n=0}^{\infty}$ converges in the $L^{1}$-norm for every $f\in L^{1}(G)$ if and only if $\mathcal{F}_{\mu}:=\{\gamma \in \Gamma:\widehat{\mu}(\gamma) =1\} $ is clopen in $\Gamma $, where $\widehat{\mu}$ is the Fourier–Stieltjes transform of $\mu $. If $\mu $ is a probability measure, then $\mathcal{F}_{\mu}$ is clopen in $\Gamma $ if and only if the closed subgroup generated by the support of $\mu $ is compact.