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Induced C *-algebras and a symmetric imprimitivity theorem. (English) Zbl 0617.22009
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 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 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.

MSC:
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
22D30 Induced representations for locally compact groups
57S25 Groups acting on specific manifolds
46L05 General theory of \(C^*\)-algebras
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