If a locally compact group G acts properly on a locally compact space X, then the induced action on C₀(X) is proper in the sense of Rieffel, with generalised fixed-point algebra C₀(GX). Rieffel's theory then gives a Morita equivalence between C₀(GX) and an ideal I in the crossed product C₀(X) x G; we identify I by describing the primitive ideals which contain it, and we deduce that I = C₀(X) x G if and only if G acts freely. We show that if a discrete group G acts on a directed graph E and every vertex of E has a finite stabiliser, then the induced action ∝ of G on the graph C* -algebra C*(E) is proper. When G acts freely on E, the generalised fixed-point algebra C*(E)∝ is isomorphic to C*(GE) and Morita equivalent to C*(E) x G, in parallel with the situation for free and proper actions on spaces, but this parallel does not seem to give useful predictions for nonfree actions.
Proceedings of the American Mathematical Society Vol. 137, Issue 7, p. 2273-2283