Projections and invariant means related to some Banach algebras

Abstract

Let $A$ be a Banach algebra with a bounded approximate identity. Our first purpose in this paper is to generalize Bekka's results for a certain class of Banach algebras. Let $G$ be an amenable locally compact topological group, and let $A$ be a left Banach $G$-module. Our second purpose, among the other things, is to define certain weak$^*$-closed subspaces of $\mathcal{B}(A,A^*)$ to consider when their weak$^*$-closed subspaces are the range of a bounded projection on $\mathcal{B}(A,A^*)$. Finally, we explore the link between the projections properties and amenability of semigroup algebras.