Induced C *-algebras and a symmetric imprimitivity theorem
Let H and K be two closed subgroups of a locally compact group G, and consider the left action of H on the right coset space G/K, and the right action of K on \(H\setminus G\). The symmetric imprimitivity theorem of \textit{M. A. Rieffel} [Math. Ann. 222, 7-22 (1976; Zbl 0328.22013)] asserts that the transformation group \(C^ *\)-algebras \(C^ *(H,G/K)\) and \(C^ *(K,H\setminus G)\) are Morita equivalent: the usual imprimitivity theorem is obtained by taking \(H=G\). When we have commuting actions of H and K on a \(C^ *\)-algebra A, there are natural actions of K and H on the induced \(C^ *\)-algebras \(Ind^ G_ H A\), \(Ind^ G_ K A\), and our main theorem says that the resulting crossed products \(C^ *(K,Ind^ G_ H A)\), \(C^ *(H,Ind^ G_ K A)\) are Morita equivalent. By making suitable choices for H and K, and passing to quotients of this equivalence, we can obtain many other known Morita equivalences for crossed products, including those occuring in \textit{P. Green}'s version of the Mackey machine [Acta Math. 140, 191-250 (1978; Zbl 0407.46053)]. We also use our results to produce a new class of examples illustrating recent work of Rosenberg and the author.
scientific article; zbMATH DE number 4000244
scientific article