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.