Fragmentability and continuity of semigroup actions

Let a topologized semigroup S act continuously and linearly on a locally convex space X. We find sufficient conditions for continuity of induced actions on the spaces of linear (compact) operators and on the dual space X^∞, for instance. The notion of fragmentability in the sense of Jayne and Rogers and its natural uniform generalizations play a major role in this paper. Our applications show that problems concerning the continuity of induced actions have satisfactory solutions for Asplund Banach spaces X (without additional restrictions, if S is a topological group) and, moreover, for a new locally convex version of Asplund spaces introduced in the paper. The starting point of this concept was the characterization of Asplund spaces due to Namioka and Phelps in terms of fragmentability.