Continuity of Actions of Groups and Semigroups on Banach Spaces

It is shown that if a locally compact group acts isometrically on a Banach space X leaving a closed subspace M invariant, and if the induced actions on M and X/M are strongly continuous, then the action on X is strongly continuous. Since this may be of interest for one-parameter semigroups, similar results are proved for actions of suitable topological semigroups. Other generalizations are given for (suitable) non-isometric actions, non-locally compact groups, and non-Banach spaces; corollaries concerning 1-cocycles and uniformly continuous actions are given.