Proper actions of groupoids on $C^*$-algebras

TL;DR

This paper extends Rieffel's concept of proper group actions on $C^*$-algebras to groupoids, establishing Morita equivalence between fixed point algebras and reduced crossed products, with examples and saturation results.

Contribution

It generalizes the notion of proper actions from groups to groupoids and proves Morita equivalence results for the associated fixed point algebras.

Findings

Generalization of proper actions to groupoids.

Morita equivalence between fixed point algebra and subalgebra of the reduced crossed product.

Saturation of proper, principal, and proper groupoid actions.

Abstract

In 1990, Rieffel defined a notion of proper action of a group $H$ on a $C^*$-algebra $A$. He then defined a generalized fixed point algebra $A^{\alpha}$ for this action and showed that $A^{\alpha}$ is Morita equivalent to an ideal of the reduced crossed product. We generalize Rieffel's notion to define proper groupoid dynamical systems and show that the generalized fixed point algebra for proper groupoid actions is Morita equivalent to a subalgebra of the reduced crossed product. We give some nontrivial examples of proper groupoid dynamical systems and show that if $(\A, G, \alpha)$ is a groupoid dynamical system such that $G$ is principal and proper, then the action of $G$ on $\A$ is saturated, that is the generalized fixed point algebra in Morita equivalent to the reduced crossed product.