Haar systems on equivalent groupoids
Author:
Dana P. Williams
Journal:
Proc. Amer. Math. Soc. Ser. B 3 (2016), 1-8
MSC (2010):
Primary 22A22; Secondary 28C15, 46L55, 46L05
DOI:
https://doi.org/10.1090/bproc/22
Published electronically:
March 28, 2016
MathSciNet review:
3478528
Full-text PDF Open Access
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Abstract | References | Similar Articles | Additional Information
Abstract: For second countable locally compact Hausdorff groupoids, the property of possessing a Haar system is preserved by equivalence.
- C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 36, L’Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683
- J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560–566. MR 97, DOI 10.2307/1968940
- Nicolas Bourbaki, Integration. II. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2004. Translated from the 1963 and 1969 French originals by Sterling K. Berberian. MR 2098271
- Rohit Dilip Holkar and Jean Renault, Hypergroupoids and $C*$-algebras, C. R. Math. Acad. Sci. Paris 351 (2013), no. 23-24, 911–914 (English, with English and French summaries). MR 3133603, DOI 10.1016/j.crma.2013.11.003
- Alexander Kumjian, Paul S. Muhly, Jean N. Renault, and Dana P. Williams, The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998), no. 5, 901–954. MR 1646047
- Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
- Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoid $C^\ast$-algebras, J. Operator Theory 17 (1987), no. 1, 3–22. MR 873460
- Paul S. Muhly and Dana P. Williams, Groupoid cohomology and the Dixmier-Douady class, Proc. London Math. Soc. (3) 71 (1995), no. 1, 109–134. MR 1327935, DOI 10.1112/plms/s3-71.1.109
- Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, vol. 170, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1724106, DOI 10.1007/978-1-4612-1774-9
- Arlan Ramsay, The Mackey-Glimm dichotomy for foliations and other Polish groupoids, J. Funct. Anal. 94 (1990), no. 2, 358–374. MR 1081649 (93a:46124)
- Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266
- Jean Renault, Représentation des produits croisés d’algèbres de groupoïdes, J. Operator Theory 18 (1987), no. 1, 67–97. MR 912813 (89g:46108)
- Jean Renault, Induced representations and hypergroupoids, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 057, 18. MR 3226993, DOI 10.3842/SIGMA.2014.057
- A. K. Seda, Haar measures for groupoids, Proc. Roy. Irish Acad. Sect. A 76 (1976), no. 5, 25–36. MR 427598
- Anthony Karel Seda, On the continuity of Haar measure on topological groupoids, Proc. Amer. Math. Soc. 96 (1986), no. 1, 115–120. MR 813822, DOI 10.1090/S0002-9939-1986-0813822-2
- Jean-Louis Tu, Non-Hausdorff groupoids, proper actions and $K$-theory, Doc. Math. 9 (2004), 565–597. MR 2117427
- Jean-Louis Tu, Ping Xu, and Camille Laurent-Gengoux, Twisted $K$-theory of differentiable stacks, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 841–910 (English, with English and French summaries). MR 2119241, DOI 10.1016/j.ansens.2004.10.002
- Dana P. Williams, Crossed products of $C{^\ast }$-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954 (2007m:46003)
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Additional Information
Dana P. Williams
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
MR Author ID:
200378
Email:
dana.williams@dartmouth.edu
Keywords:
Groupoid,
equivalent groupoids,
Haar system,
imprimitivity groupoid,
$\pi$-system,
proper action
Received by editor(s):
January 16, 2015
Received by editor(s) in revised form:
May 12, 2015
Published electronically:
March 28, 2016
Additional Notes:
This work was supported in part by a grant from the Simons Foundation.
Communicated by:
Marius Junge
Article copyright:
© Copyright 2016
by the author under
Creative Commons Attribution 3.0 License
(CC BY 3.0)