Almost periodic compactifications of transformation semigroups

In this paper we generalize the notion of (weakly) almost periodic compactification of a semitopological semigroup to the corresponding notion for transformation semigroups. The properties of these compactifications are studied and applications are made to semidirect products. 0* Introduction* Let X be a Hausdorff topological space and S a semitopological semigroup The pair (S, X) is a semitopological transformation semigroup (abbr. s.t.t.s.) if there exists a separately continuous mapping (s, x) -> sx of S x X into X (called an action) such that s(tx) -(st)x (s, teS xeX).