Irreducible representations of groupoid 𝐶*-algebras

Ionescu, Marius; Williams, Dana

doi:10.1090/S0002-9939-08-09782-7

Irreducible representations of groupoid $C^*$-algebras
HTML articles powered by AMS MathViewer

by Marius Ionescu and Dana P. Williams PDF

Proc. Amer. Math. Soc. 137 (2009), 1323-1332 Request permission

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