AN IMPRIMITIVITY THEOREM FOR REPRESENTATIONS OF LOCALLY COMPACT-GROUPS ON ARBITRARY BANACH-SPACES

We prove a general version of Mackey's Imprimitivity Theorem for induc ed representations of locally compact groups. Let G be a locally compa ct group and let H be a closed subgroup. Following Rieffel we show, us ing Morita equivalence of Banach algebras, that systems of imprimitivi ty for induction from strongly continuous Banach H-modules to strongly continuous Banach G-modules can be described in terms of an action on the induced module of C-0(G/H), the algebra of complex continuous fun ctions on G/H vanishing at infinity, which is compatible with the G-ho mogeneous structure of G/H and the strong operator topology continuity of the module action of G.