An imprimitivity theorem for representations of locally compact groups on arbitrary Banach spaces

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We prove a general version of Mackey’s Imprimitivity Theorem for induced representations of locally compact groups. Let G be a locally compact group and let H be a closed subgroup. Following Rieffel we show, using Morita equivalence of Banach algebras, that systems of imprimitivity 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 C0(G/H), the algebra of complex continuous functions on G/H vanishing at ∞, which is compatible with the G−homogeneous structure of G/H and the strong operator topology continuity of the module action of G.