SEMIGROUP ACTIONS , WEAK ALMOST PERIODICITY, AND INVARIANT MEANS

Abstract

Let S be a topological semigroup acting on a topological space X. We develop the theory of (weakly) almost periodic functions on X, with respect to S, and form the (weakly) almost periodic compactifications of X and S, with respect to each other. We then consider the notion of an action of Son a Banach space, and on its dual, and after defining S-invariant means for such a space, we give a result concerning the existence of such means, and apply it to prove the existence of a G-invariant mean on the space of weakly almost periodic functions defined on a topological space on which a topological group G acts.