General Results

Let G be a locally compact group, always assumed Hausdorff. Let H be a Banach space (which in most of our applications will be a Hubert space). A representation of G in H is a homomorphism $$ \pi :G \to GL(H) $$ of G into the group of continuous linear automorphisms of H, such that for each vector v ∈ H the map of G into H given by $$ x \to \pi (x)v $$ is continuous. One may say that the homomorphism is strongly continuous, the strong topology being the norm topology on the Banach space. [We recall here that the weak topology on H is that topology having the smallest family of open sets for which all functionals on H are continuous.]