Proper actions which are not saturated
HTML articles powered by AMS MathViewer
- by Damián Marelli and Iain Raeburn PDF
- Proc. Amer. Math. Soc. 137 (2009), 2273-2283 Request permission
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.References
- Beatriz Abadie, Generalized fixed-point algebras of certain actions on crossed products, Pacific J. Math. 171 (1995), no. 1, 1–21. MR 1362977
- Philip Green, $C^*$-algebras of transformation groups with smooth orbit space, Pacific J. Math. 72 (1977), no. 1, 71–97. MR 453917
- 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, DOI 10.1090/S0002-9939-03-07189-2
- 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
- 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, DOI 10.1017/S0143385799151940
- David Pask and Iain Raeburn, Symmetric imprimitivity theorems for graph $C^*$-algebras, Internat. J. Math. 12 (2001), no. 5, 609–623. MR 1843869, DOI 10.1142/S0129167X01000885
- 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
- 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, DOI 10.1090/cbms/103
- 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, DOI 10.1090/surv/060
- Marc A. Rieffel, Applications of strong Morita equivalence to transformation group $C^{\ast }$-algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 299–310. MR 679709
- Marc A. Rieffel, Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), no. 4, 531–562. MR 1002830
- 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
- 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, DOI 10.1016/S0723-0869(04)80002-1
- Dana P. Williams, Crossed products of $C{^\ast }$-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954, DOI 10.1090/surv/134
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
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 2273-2283
- MSC (2000): Primary 46L55
- DOI: https://doi.org/10.1090/S0002-9939-09-09867-0
- MathSciNet review: 2495260