Proper actions on imprimitivity bimodules and decompositions of Morita equivalences

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 Rieffel in 1988. 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 Rieffelʹs 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.