Decomposing the 𝐶*-algebras of groupoid extensions

Brown, Jonathan; an Huef, Astrid

doi:10.1090/S0002-9939-2014-11492-4

Decomposing the $C^*$-algebras of groupoid extensions
HTML articles powered by AMS MathViewer

by Jonathan H. Brown and Astrid an Huef PDF

Proc. Amer. Math. Soc. 142 (2014), 1261-1274 Request permission

Abstract:

We decompose the full and reduced $C^*$-algebras of an extension of a groupoid by the circle into a direct sum of twisted groupoid $C^*$-algebras.

References

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
Jonathan Henry Brown, Proper actions of groupoids on $C^*$-algebras, J. Operator Theory 67 (2012), no. 2, 437–467. MR 2928324
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
Lisa Orloff Clark and Astrid an Huef, The representation theory of $C^*$-algebras associated to groupoids, Math. Proc. Cambridge Phil. Soc. 153 (2012), 167–191.
Shaun Disney and Iain Raeburn, Homogeneous $C^\ast$-algebras whose spectra are tori, J. Austral. Math. Soc. Ser. A 38 (1985), no. 1, 9–39. MR 765447
J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472–476. MR 139135, DOI 10.1090/S0002-9939-1962-0139135-6
Geoff Goehle, Groupoid crossed products, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Dartmouth College. MR 2941279
Elliot C. Gootman, The type of some $C^{\ast }$ and $W^{\ast }$-algebras associated with transformation groups, Pacific J. Math. 48 (1973), 93–106. MR 335681
Alexander Kumjian, On $C^\ast$-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. MR 854149, DOI 10.4153/CJM-1986-048-0
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 Baruch Solel, Subalgebras of groupoid $C^*$-algebras, J. Reine Angew. Math. 402 (1989), 41–75. MR 1022793, DOI 10.1515/crll.1989.402.41
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
Milan Pahor, The structure of certain group $C^*$-algebras, Bull. Austral. Math. Soc. 47 (1993), no. 1, 169–174. MR 1201434, DOI 10.1017/S0004972700012375
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
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 (French). MR 912813
Jean Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR 2460017
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
Dana P. Williams, The structure of crossed products by smooth actions, J. Austral. Math. Soc. Ser. A 47 (1989), no. 2, 226–235. MR 1008836
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