Mansfield's imprimitivity theorem for arbitrary closed subgroups

Authors:
Astrid an Huef and Iain Raeburn

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1153-1162

MSC (2000):
Primary 46L05; Secondary 46L08, 46L55

DOI:
https://doi.org/10.1090/S0002-9939-03-07189-2

Published electronically:
August 28, 2003

MathSciNet review:
2045432

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a nondegenerate coaction of on a -algebra , and let be a closed subgroup of . The dual action is proper and saturated in the sense of Rieffel, and the generalised fixed-point algebra is the crossed product of by the homogeneous space . The resulting Morita equivalence is a version of Mansfield's imprimitivity theorem which requires neither amenability nor normality of .

**1.**K. Deicke, D. Pask, and I. Raeburn,*Coverings of directed graphs and crossed products of -algebras by coactions of homogeneous spaces*, Internat. J. Math., to appear.**2.**S. Echterhoff, S. Kaliszewski, and I. Raeburn,*Crossed products by dual coactions of groups and homogeneous spaces*, J. Operator Theory**39**(1998), 151-176. MR**99h:46124****3.**S. Echterhoff and J. Quigg,*Full duality for coactions of discrete groups*, Math. Scand.**90**(2002), 267-288. MR**2003g:46079****4.**P. Green,*The local structure of twisted covariance algebras*, Acta Math.**140**(1978), 191-250. MR**58:12376****5.**A. an Huef, I. Raeburn, and D. P. Williams,*A symmetric imprimitivity theorem for commuting proper actions*, preprint (arXiv.math.OA/0202046), 2002.**6.**S. Kaliszewski and J. Quigg,*Imprimitivity for -coactions of non-amenable groups*, Math. Proc. Cambridge Philos. Soc.**123**(1998), 101-118. MR**99a:46118****7.**M. B. Landstad, J. Phillips, I. Raeburn, and C. E. Sutherland,*Representations of crossed products by coactions and principal bundles*, Trans. Amer. Math. Soc.**299**(1987), 747-784. MR**88f:46127****8.**K. Mansfield,*Induced representations of crossed products by coactions*, J. Funct. Anal.**97**(1991), 112-161. MR**92h:46095****9.**C. K. Ng,*A remark on Mansfield's imprimitivity theorem*, Proc. Amer. Math. Soc.**126**(1998), 3767-3768. MR**99h:46111****10.**D. Olesen and G. K. Pedersen,*Applications of the Connes spectrum to -dynamical systems*, J. Funct. Anal.**30**(1978), 179-197. MR**81i:46076a****11.**D. Olesen and G. K. Pedersen,*Applications of the Connes spectrum to -dynamical systems II*, J. Funct. Anal.**36**(1980), 18-32. MR**81i:46076b****12.**J. C. Quigg,*Landstad duality for -coactions*, Math. Scand.**71**(1992), 277-294.MR**94e:46119****13.**J. C. Quigg,*Full and reduced -coactions*, Math. Proc. Cambridge Philos. Soc.**116**(1994), 435-450.MR**95g:46126****14.**J. C. Quigg and I. Raeburn,*Induced -algebras and Landstad duality for twisted coactions*, Trans. Amer. Math. Soc.**347**(1995), 2885-2915. MR**95j:46080****15.**I. Raeburn,*On crossed products by coactions and their representation theory*, Proc. London Math. Soc.**64**(1992), 625-652. MR**93e:46080****16.**M. A. Rieffel,*Proper actions of groups on -algebras*, Mappings of Operator Algebras, Progr. Math., vol. 84, Birkhäuser, Boston, 1988, pp. 141-182. MR**92i:46079****17.**M. A. Rieffel,*Integrable and proper actions on -algebras, and square integrable representations of groups*, preprint (arXiv.math.OA/9809098), 1999.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46L05,
46L08,
46L55

Retrieve articles in all journals with MSC (2000): 46L05, 46L08, 46L55

Additional Information

**Astrid an Huef**

Affiliation:
School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia

Email:
astrid@maths.unsw.edu.au

**Iain Raeburn**

Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Australia

Email:
iain@frey.newcastle.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-03-07189-2

Received by editor(s):
June 30, 2002

Received by editor(s) in revised form:
December 18, 2002

Published electronically:
August 28, 2003

Additional Notes:
This research was supported by grants from the Australian Research Council, the University of New South Wales and the University of Newcastle.

Communicated by:
David R. Larson

Article copyright:
© Copyright 2003
American Mathematical Society