Module Homomorphisms Associated with Banach Algebras

Abstract

Let $A$ be a Banach algebra. In this paper, among the other things, we present a few results in the theory of homomorphisms on $A^*$. We want to find out when the equality $T(af) = aT(f)$ for every $a \in A$ and $f \in A^*$ implies the equality $T(Ff) = FT(f)$ for every $f \in A^*$ and $F \in A^{**}$. One of the main results of this paper is to introduce and study the notion of a weakly almost periodic operator.