A contribution to group representations in locally convex spaces

Let U be a continuous representation of a (connected) locally compact group G in a separated locally convex space E. It is shown that the study of U is equivalent to the study of a family U _i of continuous representations of G in Fréchet spaces F _i. If U is equicontinuous, the F _i are Banach spaces, and the U _i are realized by isometric operators. When U is topologically irreducible, it is Naïmark equivalent to a Fréchet (or isometric Banach, in the equicontinuous case) continuous representation. Similar results hold for semi-groups.