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

TL;DR

This paper explores the functorial properties of fixed-point algebras in nonabelian duality, establishing naturality results and a categorical Landstad duality for crossed products, advancing the understanding of noncommutative dynamical systems.

Contribution

It proves the functoriality of fixed-point algebra assignment and the naturality of Rieffel's Morita equivalence, leading to a categorical Landstad duality in nonabelian duality.

Findings

Fixed-point algebra assignment is functorial.

Rieffel's Morita equivalence is natural.

Categorical Landstad duality characterizes crossed products.

Abstract

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let gamma be the induced action on C_0(X). We consider a category in which the objects are C*-dynamical systems (A, G, alpha) for which there is an equivariant homomorphism of (C_0(X), gamma) 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^alpha which is Morita equivalent to A times_{alpha,r} G. We show that the assignment (A, alpha) maps to A^alpha 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.