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Principal groupoid  -algebras with bounded trace
Authors:
Lisa Orloff Clark and Astrid an Huef
Journal:
Proc. Amer. Math. Soc. 136 (2008), 623-634
MSC (2000):
Primary 46L05, 46L55
Published electronically:
October 26, 2007
MathSciNet review:
2358504
Full-text PDF Free Access
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Additional Information
Abstract: Suppose  is a second countable, locally compact, Hausdorff, principal groupoid with a fixed left Haar system. We define a notion of integrability for groupoids and show  is integrable if and only if the groupoid  -algebra  has bounded trace.
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- 1.
- R.J. Archbold and K. Deicke, Bounded trace
 -algebras and integrable actions, Math. Zeit. 250 (2005), 393-410. MR 2178791 (2006i:46097)
- 2.
- R.J. Archbold and A. an Huef, Strength of convergence in the orbit space of a transformation group, J. Funct. Anal. 235 (2006), 90-121. MR 2216441 (2007a:22004)
- 3.
- R.J. Archbold and D.W.B. Somerset, Transition probabilities and trace functions for
 -algebras, Math. Scand. 73 (1993), 81-111. MR 1251700 (95f:46095)
- 4.
- L.O. Clark, Classifying the type of principal groupoid -algebras, J. Operator Theory, 57 (2007). 251-266.
- 5.
- P. Green,
 -algebras of transformation groups with smooth orbit space, Pacific J. Math. 72 (1977), 71-97. MR 0453917 (56:12170)
- 6.
- A. an Huef, Integrable actions and the transformation groups whose
 -algebras have bounded trace, Indiana Univ. Math. J. 51 (2002), 1197-1233. MR 1947873 (2004b:46095)
- 7.
- P.S. Muhly, J. Renault, and D. P. Williams, Equivalence and isomorphism for groupoid -algebras, J. Operator Theory 17 (1987), 3-22. MR 873460 (88h:46123)
- 8.
- P.S. Muhly and D.P. Williams, Continuous trace groupoid
 -algebras, Math. Scand. 66 (1990), 231-241. MR 1075140 (91j:46081)
- 9.
- G.K. Pedersen,
 -algebras and their automorphism groups, Academic Press, London, 1979. MR 548006 (81e:46037)
- 10.
- G.K. Pedersen, Analysis Now, Springer-Verlag, New York, 1989. MR 971256 (90f:46001)
- 11.
- A. Ramsay, The Mackey-Glimm dichotomy for foliations and other Polish groupoids, J. Funct. Anal. 94 (1990), 358-374. MR 1081649 (93a:46124)
- 12.
- J. Renault, A groupoid approach to
 -algebras, Lecture Notes in Mathematics, No. 793, Springer-Verlag, New York, 1980. MR 584266 (82h:46075)
- 13.
- M.A. Rieffel, Integrable and proper actions on
 -algebras, and square-integrable representations of groups, Expo. Math. 22 (2004), 1-53. MR 2166968 (2006g:46108)
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Additional Information
Lisa Orloff Clark
Affiliation:
Department of Mathematical Sciences, Susquehanna University, Selinsgrove, Pennsylvania 17870
Email:
clarklisa@susqu.edu
Astrid an Huef
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Email:
astrid@unsw.edu.au
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-09035-1
PII:
S 0002-9939(07)09035-1
Keywords:
Locally compact groupoid,
$C^*$-algebra,
bounded trace
Received by editor(s):
August 23, 2006
Received by editor(s) in revised form:
December 6, 2006
Published electronically:
October 26, 2007
Additional Notes:
This research was supported by the Australian Research Council and an AWM-NSF Mentoring Travel Grant.
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2007
American Mathematical Society
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