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Non-Hausdorff étale groupoids


Author: R. Exel
Journal: Proc. Amer. Math. Soc. 139 (2011), 897-907
MSC (2010): Primary 46L55; Secondary 22A22
DOI: https://doi.org/10.1090/S0002-9939-2010-10477-X
Published electronically: October 28, 2010
MathSciNet review: 2745642
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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.


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  • 1. C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographie de l'Enseignement Mathématique, 36, Genève, 2000. MR 1799683 (2001m:22005)
  • 2. A. Connes, A survey of foliations and operator algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., 38, Amer. Math. Soc., 1982, 521-628. MR 679730 (84m:58140)
  • 3. J. Cuntz and W. Krieger, A class of $ C^*$-algebras and topological Markov chains, Inventiones Math., 56 (1980), 251-268. MR 561974 (82f:46073a)
  • 4. R. Exel, Inverse semigroups and combinatorial C*-algebras, Bull. Braz. Math. Soc. (N.S.), 39 (2008), 191-313. MR 2419901 (2009b:46115)
  • 5. R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math., 512 (1999), 119-172. MR 1703078 (2000i:46064)
  • 6. R. Exel, M. Laca and J. Quigg, Partial dynamical systems and $ C^*$-algebras generated by partial isometries, J. Operator Theory, 47 (2002), 169-186. MR 1905819 (2003f:46108)
  • 7. A. Kumjian, On C*-diagonals, Canad. J. Math., 38 (1986), no. 4, 969-1008. MR 854149 (88a:46060)
  • 8. A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math., 184 (1998), 161-174. MR 1626528 (99i:46049)
  • 9. A. L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Birkhäuser, 1999. MR 1724106 (2001a:22003)
  • 10. J. Renault, A groupoid approach to $ C^*$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, 1980. MR 584266 (82h:46075)
  • 11. J. Renault, Cartan subalgebras in $ C^*$-algebras, Irish Math. Soc. Bull., 61 (2008), 29-63. MR 2460017 (2009k:46135)
  • 12. J. Renault, The ideal structure of groupoid crossed product $ C^*$-algebras [with an appendix by G. Skandalis], J. Operator Theory, 25 (1991), 3-36. MR 1191252 (94g:46074)

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Additional Information

R. Exel
Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, Brazil
Email: r@exel.com.br

DOI: https://doi.org/10.1090/S0002-9939-2010-10477-X
Keywords: Non-Hausdorff groupoids, essentially principal groupoids
Received by editor(s): July 30, 2009
Published electronically: October 28, 2010
Additional Notes: The author was partially supported by CNPq.
Communicated by: Marius Junge
Article copyright: © Copyright 2010 Ruy Exel

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