ar X iv : 0 80 1 . 01 61 v 3 [ m at h . O A ] 1 4 A pr 2 00 8 PROPER ACTIONS , FIXED - POINT ALGEBRAS AND NATURALITY IN NONABELIAN DUALITY

Suppose a locally compact groupG 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 which is Morita equivalent to A×α,rG. We show that the assignment (A,α) 7→ A 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.