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Proper actions on imprimitivity bimodules and decompositions of Morita equivalences. (English) Zbl 1031.46068
Summary: We consider a class of proper actions of locally compact groups on imprimitivity bimodules over \(C^*\)-algebras which behave like the proper actions on \(C^*\)-algebras introduced by M. A. Rieffel [Prog. Math. 84, 141-182 (1991; Zbl 0749.22003)]. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel’s theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different proofs of the symmetric imprimitivity theorem for actions on graph algebras yield isomorphic equivalences, and this gives new information about the amenability of actions on graph algebras.

46L08 \(C^*\)-modules
46L05 General theory of \(C^*\)-algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
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