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Proper actions which are not saturated

Authors: Damián Marelli and Iain Raeburn
Journal: Proc. Amer. Math. Soc. 137 (2009), 2273-2283
MSC (2000): Primary 46L55
Published electronically: March 11, 2009
MathSciNet review: 2495260
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Abstract: If a locally compact group $ G$ acts properly on a locally compact space $ X$, then the induced action on $ C_0(X)$ is proper in the sense of Rieffel, with generalised fixed-point algebra $ C_0(G\backslash X)$. Rieffel's theory then gives a Morita equivalence between $ C_0(G\backslash X)$ and an ideal $ I$ in the crossed product $ C_0(X)\times G$; we identify $ I$ by describing the primitive ideals which contain it, and we deduce that $ I=C_0(X)\times 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 $ \alpha$ of $ G$ on the graph $ C^*$-algebra $ C^*(E)$ is proper. When $ G$ acts freely on $ E$, the generalised fixed-point algebra $ C^*(E)^\alpha$ is isomorphic to $ C^*(G\backslash E)$ and Morita equivalent to $ C^*(E)\times 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.

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Additional Information

Damián Marelli
Affiliation: ARC Centre for Complex Dynamic Systems and Control, University of Newcastle, NSW 2308, Australia

Iain Raeburn
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia

Received by editor(s): February 11, 2008
Published electronically: March 11, 2009
Additional Notes: This research was supported by the Australian Research Council through the ARC Centre for Complex Dynamic Systems and Control.
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society

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