Naturality of Rieffel\u27s Morita equivalence for proper actions

Suppose that a locally compact group G acts freely and properly on the right of a locally compact space T . Rieffel proved that if is an action of G on a C*-algebra A and there is an equivariant embedding of C0(T ) in M(A), then the action α of G on A is proper, and the crossed product A⋊α,rG is Morita equivalent to a generalised fixed-point algebra Fix(A,α ) in M(A)α. We show that the assignment (A,α) → Fix(A,α ) extends to a functor Fix 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 Fix. 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