Naturality of Rieffel's Morita equivalence for proper actions

Abstract

Suppose that a locally compact group $G$ acts freely and properly on the right of a locally compact space $T$. Rieffel proved that if $\alpha$ 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 $\alpha$ of $G$ on $A$ is proper, and the crossed product $A\rtimes_{\alpha,r}G$ is Morita equivalent to a generalised fixed-point algebra $\Fix(A,\alpha)$ in $M(A)^\alpha$. We show that the assignment $(A,\alpha)\mapsto\Fix(A,\alpha)$ 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.