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 acts properly on a locally compact space , then the induced action on is proper in the sense of Rieffel, with generalised fixed-point algebra . Rieffel's theory then gives a Morita equivalence between and an ideal in the crossed product ; we identify by describing the primitive ideals which contain it, and we deduce that if and only if acts freely. We show that if a discrete group acts on a directed graph and every vertex of has a finite stabiliser, then the induced action of on the graph -algebra is proper. When acts freely on , the generalised fixed-point algebra is isomorphic to and Morita equivalent to , 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

Email:
damian.marelli@newcastle.edu.au

**Iain Raeburn**

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

Email:
raeburn@uow.edu.au

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-09867-0

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