Proper actions which are not saturated

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(G\X). Rieffel’s theory then gives a Morita equivalence between C0(G\X) and an ideal I in the crossed product C0(X) × G; we identify I by describing the primitive ideals which contain it, and we deduce that I = C0(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 α of G on the graph C∗-algebra C∗(E) is proper. When G acts freely on E, the generalised fixed-point algebra C∗(E)α is isomorphic to C∗(G\E) and Morita equivalent to C∗(E) × 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.