Irreducible representations of groupoid $C^*$-algebras
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- by Marius Ionescu and Dana P. Williams
- Proc. Amer. Math. Soc. 137 (2009), 1323-1332
- DOI: https://doi.org/10.1090/S0002-9939-08-09782-7
- Published electronically: December 4, 2008
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Abstract:
If $G$ is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.References
- Lisa Orloff Clark, CCR and GCR groupoid $C^*$-algebras, Indiana Univ. Math. J. 56 (2007), no. 5, 2087–2110. MR 2359724, DOI 10.1512/iumj.2007.56.2955
- A. Connes, A survey of foliations and operator algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 521–628. MR 679730
- Valentin Deaconu, Groupoids associated with endomorphisms, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1779–1786. MR 1233967, DOI 10.1090/S0002-9947-1995-1233967-5
- T. Fack and G. Skandalis, Sur les représentations et idéaux de la $C^{\ast }$-algèbre d’un feuilletage, J. Operator Theory 8 (1982), no. 1, 95–129 (French). MR 670180
- James Glimm, Families of induced representations, Pacific J. Math. 12 (1962), 885–911. MR 146297, DOI 10.2140/pjm.1962.12.885
- André Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1984), 70–97 (French). Transversal structure of foliations (Toulouse, 1982). MR 755163
- 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, DOI 10.1353/ajm.1998.0040
- E. C. Lance, Hilbert $C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694, DOI 10.1017/CBO9780511526206
- George W. Mackey, Imprimitivity for representations of locally compact groups. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 537–545. MR 31489, DOI 10.1073/pnas.35.9.537
- 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, Jean N. Renault, and Dana P. Williams, Continuous-trace groupoid $C^\ast$-algebras. III, Trans. Amer. Math. Soc. 348 (1996), no. 9, 3621–3641. MR 1348867, DOI 10.1090/S0002-9947-96-01610-8
- Paul S. Muhly and Dana P. Williams, Continuous trace groupoid $C^*$-algebras, Math. Scand. 66 (1990), no. 2, 231–241. MR 1075140, DOI 10.7146/math.scand.a-12307
- Paul S. Muhly and Dana P. Williams, Continuous trace groupoid $C^*$-algebras. II, Math. Scand. 70 (1992), no. 1, 127–145. MR 1174207, DOI 10.7146/math.scand.a-12390
- Paul S. Muhly and Dana P. Williams, Renault’s equivalence theorem for groupoid crossed products, NYJM Monographs, vol. 3, State University of New York University at Albany, Albany, NY, 2008. Available at http://nyjm.albany.edu:8000/m/2008/3.htm.
- 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
- Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
- Jean Renault, Représentation des produits croisés d’algèbres de groupoïdes, J. Operator Theory 18 (1987), no. 1, 67–97 (French). MR 912813
- Marc A. Rieffel, Induced representations of $C^{\ast }$-algebras, Advances in Math. 13 (1974), 176–257. MR 353003, DOI 10.1016/0001-8708(74)90068-1
- Dana P. Williams, The topology on the primitive ideal space of transformation group $C^{\ast }$-algebras and C.C.R. transformation group $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 266 (1981), no. 2, 335–359. MR 617538, DOI 10.1090/S0002-9947-1981-0617538-7
- 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
- H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75. MR 739904, DOI 10.1007/BF02329732
Bibliographic Information
- Marius Ionescu
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
- Address at time of publication: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
- Email: ionescu@math.uconn.edu
- Dana P. Williams
- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
- MR Author ID: 200378
- Email: dana.p.williams@Dartmouth.edu
- Received by editor(s): October 19, 2007
- Published electronically: December 4, 2008
- Communicated by: Marius Junge
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1323-1332
- MSC (2000): Primary 46L55, 46L05; Secondary 22A22
- DOI: https://doi.org/10.1090/S0002-9939-08-09782-7
- MathSciNet review: 2465655