Continuous-trace groupoid crossed products

Fulman, Igor; Muhly, Paul; Williams, Dana

doi:10.1090/S0002-9939-03-07158-2

Continuous-trace groupoid crossed products
HTML articles powered by AMS MathViewer

by Igor Fulman, Paul S. Muhly and Dana P. Williams PDF

Proc. Amer. Math. Soc. 132 (2004), 707-717 Request permission

Abstract:

Let $G$ be a second countable, locally compact groupoid with Haar system, and let $\mathcal {A}$ be a bundle of $C^{\ast }$-algebras defined over the unit space of $G$ on which $G$ acts continuously. We determine conditions under which the associated crossed product $C^{\ast }(G;\mathcal {A})$ is a continuous trace $C^{\ast }$-algebra.

References

P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard, and M. Vergne, Représentations des groupes de Lie résolubles, Monographies de la Société Mathématique de France, No. 4, Dunod, Paris, 1972. MR 0444836
Klaus Deicke, Pointwise unitary coactions on $C^*$-algebras with continuous trace, J. Operator Theory 43 (2000), no. 2, 295–327. MR 1753413
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
Philip Green, $C^*$-algebras of transformation groups with smooth orbit space, Pacific J. Math. 72 (1977), no. 1, 71–97. MR 453917, DOI 10.2140/pjm.1977.72.71
Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. MR 493349, DOI 10.1007/BF02392308
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
Ru Ying Lee, On the $C^*$-algebras of operator fields, Indiana Univ. Math. J. 25 (1976), no. 4, 303–314. MR 410400, DOI 10.1512/iumj.1976.25.25026
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, 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
May Nilsen, $C^*$-bundles and $C_0(X)$-algebras, Indiana Univ. Math. J. 45 (1996), no. 2, 463–477. MR 1414338, DOI 10.1512/iumj.1996.45.1086
Dorte Olesen and Iain Raeburn, Pointwise unitary automorphism groups, J. Funct. Anal. 93 (1990), no. 2, 278–309. MR 1073288, DOI 10.1016/0022-1236(90)90130-D
Iain Raeburn and Jonathan Rosenberg, Crossed products of continuous-trace $C^\ast$-algebras by smooth actions, Trans. Amer. Math. Soc. 305 (1988), no. 1, 1–45. MR 920145, DOI 10.1090/S0002-9947-1988-0920145-6
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, Représentation des produits croisés d’algèbres de groupoïdes, J. Operator Theory 18 (1987), no. 1, 67–97 (French). MR 912813
Masamichi Takesaki, A liminal crosed product of a uniformly hyperfinite $C^{\ast }$-algebra by a compact Abelian automorphism group, J. Functional Analysis 7 (1971), 140–146. MR 0275184, DOI 10.1016/0022-1236(71)90049-8
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