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

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

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.

**1.**Beatriz Abadie,*Generalized fixed-point algebras of certain actions on crossed products*, Pacific J. Math.**171**(1995), no. 1, 1–21. MR**1362977****2.**Philip Green,*𝐶*-algebras of transformation groups with smooth orbit space*, Pacific J. Math.**72**(1977), no. 1, 71–97. MR**0453917****3.**Astrid An Huef and Iain Raeburn,*Mansfield’s imprimitivity theorem for arbitrary closed subgroups*, Proc. Amer. Math. Soc.**132**(2004), no. 4, 1153–1162. MR**2045432**, https://doi.org/10.1090/S0002-9939-03-07189-2**4.**S. Kaliszewski, John Quigg, and Iain Raeburn,*Skew products and crossed products by coactions*, J. Operator Theory**46**(2001), no. 2, 411–433. MR**1870415****5.**Alex Kumjian and David Pask,*𝐶*-algebras of directed graphs and group actions*, Ergodic Theory Dynam. Systems**19**(1999), no. 6, 1503–1519. MR**1738948**, https://doi.org/10.1017/S0143385799151940**6.**David Pask and Iain Raeburn,*Symmetric imprimitivity theorems for graph 𝐶*-algebras*, Internat. J. Math.**12**(2001), no. 5, 609–623. MR**1843869**, https://doi.org/10.1142/S0129167X01000885**7.**N. Christopher Phillips,*Equivariant 𝐾-theory for proper actions*, Pitman Research Notes in Mathematics Series, vol. 178, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR**991566****8.**Iain Raeburn,*Graph algebras*, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. MR**2135030****9.**Iain Raeburn and Dana P. Williams,*Morita equivalence and continuous-trace 𝐶*-algebras*, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. MR**1634408****10.**Marc A. Rieffel,*Applications of strong Morita equivalence to transformation group 𝐶*-algebras*, Operator algebras and applications, Part I (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 299–310. MR**679709****11.**Marc A. Rieffel,*Deformation quantization of Heisenberg manifolds*, Comm. Math. Phys.**122**(1989), no. 4, 531–562. MR**1002830****12.**Marc A. Rieffel,*Proper actions of groups on 𝐶*-algebras*, Mappings of operator algebras (Philadelphia, PA, 1988) Progr. Math., vol. 84, Birkhäuser Boston, Boston, MA, 1990, pp. 141–182. MR**1103376****13.**Marc A. Rieffel,*Integrable and proper actions on 𝐶*-algebras, and square-integrable representations of groups*, Expo. Math.**22**(2004), no. 1, 1–53. MR**2166968**, https://doi.org/10.1016/S0723-0869(04)80002-1**14.**Dana P. Williams,*Crossed products of 𝐶*-algebras*, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR**2288954**

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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:
https://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