Rocky Mountain Journal of Mathematics

Inverse semigroup actions on groupoids

Alcides Buss and Ralf Meyer

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We define inverse semigroup actions on topological groupoids by partial equivalences. From such actions, we construct saturated Fell bundles over inverse semigroups and non-Hausdorff \'etale groupoids. We interpret these as actions on $C^*$\nobreakdash -algebras by Hilbert bimodules and describe the section algebras of these Fell bundles.

Our constructions give saturated Fell bundles over non-Hausdorff \'etale groupoids that model actions on locally Hausdorff spaces. We show that these Fell bundles are usually not Morita equivalent to an action by automorphisms, that is, the Packer-Raeburn stabilization trick does not generalize to non-Hausdorff groupoids.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 1 (2017), 53-159.

Dates
First available in Project Euclid: 3 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1488531894

Digital Object Identifier
doi:10.1216/RMJ-2017-47-1-53

Mathematical Reviews number (MathSciNet)
MR3619758

Zentralblatt MATH identifier
06702339

Subjects
Primary: 20M18: Inverse semigroups 22A22: Topological groupoids (including differentiable and Lie groupoids) [See also 58H05] 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

Keywords
Inverse semigroups groupoids actions partial equivalences Fell bundles stabilization trick

Citation

Buss, Alcides; Meyer, Ralf. Inverse semigroup actions on groupoids. Rocky Mountain J. Math. 47 (2017), no. 1, 53--159. doi:10.1216/RMJ-2017-47-1-53. https://projecteuclid.org/euclid.rmjm/1488531894


Export citation

References

  • Jean Bénabou, Introduction to bicategories, Springer, Berlin, 1967.
  • Nicolas Bourbaki, Topologie générale, Elem. Math., Hermann, Paris, 1971.
  • Alcides Buss and Ruy Exel, Twisted actions and regular Fell bundles over inverse semigroups, Proc. Lond. Math. Soc. 103 (2011), 235–270.
  • ––––, Fell bundles over inverse semigroups and twisted étale groupoids, J. Op. Th. 67 (2012), 153–205.
  • Alcides Buss and Ralf Meyer, Crossed products for actions of crossed modules on $\textup C^*$ -algebras, J. Noncomm. Geom., accepted.
  • Alcides Buss, Ralf Meyer and Chenchang Zhu, A higher category approach to twisted actions on $\textup C^*$ -algebras, Proc. Edinb. Math. Soc. 56 (2013), 387–426.
  • Jérôme Chabert and Siegfried Echterhoff, Twisted equivariant $KK$-theory and the Baum-Connes conjecture for group extensions, $K$-Theory 23 (2001), 157–200.
  • Lisa Orloff Clark, Astrid an Huef and Iain Raeburn, The equivalence relations of local homeomorphisms and Fell algebras, New York J. Math. 19 (2013), 367–394.
  • Valentin Deaconu, Groupoids associated with endomorphisms, Trans. Amer. Math. Soc. 347 (1995), 1779–1786.
  • Claire Debord, Holonomy groupoids of singular foliations, J. Diff. Geom. 58 (2001), 467–500.
  • Ruy Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), 3481–3494.
  • ––––, Inverse semigroups and combinatorial $C^*$ -algebras, Bull. Brazilian Math. Soc. 39 (2008), 191–313.
  • ––––, Noncommutative Cartan subalgebras of $C^*$ -algebras, New York J. Math. 17 (2011), 331–382.
  • Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), 191–250.
  • Rohit Dilip Holkar, Topological construction of $\textup {C}^*$ -correspondences for groupoid $\textup {C}^*$-algebras, Ph.D dissertation, Georg-August-Universität, Göttingen, 2014.
  • Gennadi G. Kasparov and Georges Skandalis, Groups acting on buildings, operator $K$-theory, and Novikov's conjecture, $K$-Theory 4 (1991), 303–337.
  • Alex Kumjian, Fell bundles over groupoids, Proc. Amer. Math. Soc. 126 (1998), 1115–1125.
  • Mark V. Lawson, Inverse semigroups: The theory of partial symmetries, World Scientific Publishing Co., River Edge, NJ, 1998.
  • Tom Leinster, Basic bicategories, 1998..
  • Ralf Meyer and Ryszard Nest, $C^*$-algebras over topological spaces: The bootstrap class, Munster J. Math. 2 (2009), 215–252.
  • Ralf Meyer and Chenchang Zhu, Groupoids in categories with pretopology, Theory Appl. Categ. 30 (2015), 1906–1998.
  • Ieke Moerdijk, Orbifolds as groupoids: An introduction, in Orbifolds in mathematics and physics, Contemp. Math. 310, American Mathematical Society Providence, 2002.
  • Paul S. Muhly, Jean N. Renault and Dana P. Williams, Equivalence and isomorphism for groupoid $C^*$-algebras, J. Op. Th. 17 (1987), 3–22.
  • Paul S. Muhly and Dana P. Williams, Equivalence and disintegration theorems for Fell bundles and their $C^*$ -algebras, Disser. Math. 456 (2008), 1–57.
  • Paul S. Muhly and Dana P. Williams, Renault's equivalence theorem for groupoid crossed products, New York J. Math. Mono. 3 Albany, 2008.
  • May Nilsen, $C^*$ -bundles and $C_0(X)$-algebras, Indiana Univ. Math. J. 45 (1996), 463–477.
  • Radu Popescu, Equivariant $E$-theory for groupoids acting on $C^*$ -algebras, J. Funct. Anal. 209 (2004), 247–292.
  • John Quigg and Nándor Sieben, $C^*$ -actions of $r$ -discrete groupoids and inverse semigroups, J. Australian Math. Soc. 66 (1999), 143–167.
  • Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace $C^*$ -algebras, Math. Surv. Mono. 60, American Mathematical Society, Providence, RI, 1998.
  • Jean Renault, A groupoid approach to $\textup C^*$ -algebras, Lect. Notes Math. 793, Springer, Berlin, 1980.
  • ––––, Représentation des produits croisés d'algèbres de groupoï des, J. Op. Th. 18 (1987), 67–97.
  • ––––, Cartan subalgebras in $C^*$ -algebras, Irish Math. Soc. Bull. 61 (2008), 29–63.
  • Nándor Sieben, $C^*$ -crossed products by partial actions and actions of inverse semigroups, J. Australian Math. Soc. 63 (1997), 32–46.
  • ––––, $C^*$ -crossed products by twisted inverse semigroup actions, J. Op. Th. 39 (1998), 361–393.
  • Giorgio Trentinaglia, Tannaka duality for proper Lie groupoids, Ph.D. dissertation, Utrecht University, Utrecht, 2008.
  • Jean-Louis Tu, Non-Hausdorff groupoids, proper actions and $K$ -theory, Doc. Math. 9 (2004), 565–597.
  • Alan Weinstein, Linearization of regular proper groupoids, J. Inst. Math. Jussieu 1 (2002), 493–511.
  • Nguyen Tien Zung, Proper groupoids and momentum maps: Linearization, affinity, and convexity, Ann. Sci. Ecole Norm. 39 (2006), 841–869.