Rieffel proper actions

TL;DR

This paper characterizes Rieffel proper actions of locally compact groups on C*-algebras, providing new examples and counterexamples that deepen understanding of their structure and fixed-point algebras.

Contribution

It offers a simple characterization of Rieffel proper actions and presents novel examples illustrating the diversity of such actions and their fixed-point algebras.

Findings

Counterexamples where properness isn't induced by a nondegenerate equivariant *-homomorphism

Examples of actions with multiple Rieffel proper structures

Clarification of the relationship between proper actions and fixed-point algebras

Abstract

In the late 1980's Marc Rieffel introduced a notion of properness for actions of locally compact groups on C*-algebras which, among other things, allows the construction of generalised fixed-point algebras for such actions. In this paper we give a simple characterisation of Rieffel proper actions and use this to obtain several (counter) examples for the theory. In particular, we provide examples of Rieffel proper actions $\alpha:G\to\mathrm{Aut}(A)$ for which properness is not induced by a nondegenerate equivariant *-homomorphism $\phi:C_0(X)\to \mathcal{M}(A)$ for any proper $G$-space $X$. Other examples, based on earlier work of Meyer, show that a given action might carry different structures for Rieffel properness with different generalised fixed-point algebras.