\(A\)-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.