On $\phi$-ergodic property of Banach modules

Abstract

Let ${\mathcal A}$ be a Banach algebra and let $\phi$ be a non-zero character on ${\mathcal A}$. We introduce the notion of $\phi$-ergodic property for a Banach right ${\mathcal A}$-module $X$. This concept considerably generalizes the existence of $\phi$-means of norm one on ${\mathcal A}^*$. We also show that the $\phi$-ergodic property of $X$ is related to some other properties such as a Hahn-Banach type extension property and the existence of $\phi$-means of norm one on a certain subspace of ${\mathcal A}^*$. Finally, we give some characterizations for $\phi$-amenability of a Banach algebra in terms of its closed ideals.