Non-Hausdorff étale groupoids

Exel, R.

doi:10.1090/S0002-9939-2010-10477-X

Non-Hausdorff étale groupoids
HTML articles powered by AMS MathViewer

by R. Exel PDF

Proc. Amer. Math. Soc. 139 (2011), 897-907

Abstract:

We present examples of non-Hausdorff, étale, essentially principal groupoids for which three results, known to hold in the Hausdorff case, fail. These results are: (A) the subalgebra of continuous functions on the unit space is maximal abelian within the reduced groupoid C*-algebra, (B) every nonzero ideal of the reduced groupoid C*-algebra has a nonzero intersection with the subalgebra of continuous functions on the unit space, and (C) the open support of a normalizer is a bisection.

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
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
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Ruy Exel, Inverse semigroups and combinatorial $C^\ast$-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191–313. MR 2419901, DOI 10.1007/s00574-008-0080-7
Ruy Exel and Marcelo Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119–172. MR 1703078, DOI 10.1515/crll.1999.051
Ruy Exel, Marcelo Laca, and John Quigg, Partial dynamical systems and $C^*$-algebras generated by partial isometries, J. Operator Theory 47 (2002), no. 1, 169–186. MR 1905819
Alexander Kumjian, On $C^\ast$-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. MR 854149, DOI 10.4153/CJM-1986-048-0
Alex Kumjian, David Pask, and Iain Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. MR 1626528, DOI 10.2140/pjm.1998.184.161
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
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266
Jean Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR 2460017
Jean Renault, The ideal structure of groupoid crossed product $C^\ast$-algebras, J. Operator Theory 25 (1991), no. 1, 3–36. With an appendix by Georges Skandalis. MR 1191252