An imprimitivity theorem for representations of locally compact groups on arbitrary Banach spaces (Q1366634)

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 \(C_0(G/H)\), the algebra of complex continuous functions on \(G/H\) vanishing at \(\infty\), which is compatible with the \(G\)-homogeneous structure of \(G/H\) and the strong operator topology continuity of the module action of \(G\).