Proper actions, fixed-point algebras and naturality in nonabelian duality

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C₀ (X). We consider a category in which the objects are C^*-dynamical systems (A, G, α) for which there is an equivariant homomorphism of (C₀ (X), γ) into the multiplier algebra M (A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra A^α which is Morita equivalent to A ×_{α, r} G. We show that the assignment (A, α) {mapping} A^α is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.