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Suppose that a locally compact group G acts freely and properly on the right of a locally compact space T. Rieffel proved that if α is an action of G on a C *-algebra A and there is an equivariant embedding of C 0(T) in M(A), then the action α of G on A is proper, and the crossed product  is Morita equivalent to a generalised fixed-point algebra  in M(A) α . We show that the assignment  extends to a functor  on a category of C *-dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel’s Morita equivalence implements a natural isomorphism between a crossed-product functor and . From this, we deduce naturality of Mansfield imprimitivity for crossed products by coactions, improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg-Raeburn, and naturality of a Morita equivalence for graph algebras due to Kumjian and Pask.

Naturality of Rieffel’s Morita Equivalence for Proper Actions