Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T19:20:13.330Z Has data issue: false hasContentIssue false
  • English
  • Français

INVARIANT SETS AND NORMAL SUBGROUPOIDS OF UNIVERSAL ÉTALE GROUPOIDS INDUCED BY CONGRUENCES OF INVERSE SEMIGROUPS

Part of: Selfadjoint operator algebras Topological and differentiable algebraic systems Semigroups

Published online by Cambridge University Press:  13 April 2021

FUYUTA KOMURA Open the ORCID record for FUYUTA KOMURA [Opens in a new window]
FUYUTA KOMURA*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan
*
e-mail: fuyuta.k@keio.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.

Keywords

C*-algebrasétale groupoids

MSC classification

Primary: 20M18: Inverse semigroups
Secondary: 22A22: Topological groupoids (including differentiable and Lie groupoids) 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 1 , August 2022 , pp. 99 - 118
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Australian Mathematical Publishing Association Inc. 2021

Footnotes

Communicated by Aidan Sims

This work was supported by JSPS KAKENHI 20J10088.

References

Komura, F., ‘Quotients of étale groupoids and the abelianizations of groupoid C*-algebras’, J. Aust. Math. Soc., to appear. Published online (7 April 2020).10.1017/S1446788720000154CrossRefGoogle Scholar
LaLonde, S. M. and Milan, D., ‘Amenability and uniqueness for groupoids associated with inverse semigroups’, Semigroup Forum 95(2) (2017), 321–344.10.1007/s00233-016-9839-0CrossRefGoogle Scholar
Lawson, M. V., Inverse Semigroups (World Scientific, Singapore, 1998).10.1142/3645CrossRefGoogle Scholar
Paterson, A., Groupoids, Inverse Semigroups, and their Operator Algebras , Progress in Mathematics, 170 (Birkhäuser, Boston, MA, 2012).Google Scholar
Petrich, M., Inverse Semigroups , Pure and Applied Mathematics (Wiley, New York, 1984).Google Scholar
Piochi, B., ‘Solvability in inverse semigroups’, Semigroup Forum 34(1) (1986), 287–303.10.1007/BF02573169CrossRefGoogle Scholar
Renault, J., A Groupoid Approach to C*-Algebras , Lecture Notes in Mathematics, 793 (Springer, Berlin, 1980).CrossRefGoogle Scholar
Sims, A., ‘Hausdorff étale groupoids and their C*-algebras’, in: Operator Algebras and Dynamics: Groupoids, Crossed Products, and Rokhlin Dimension , Advanced Courses in Mathematics, CRM Barcelona (ed. Perera, F.) (Birkhäuser, Cham, 2020).10.1007/978-3-030-39713-5CrossRefGoogle Scholar
Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, Adv. Math. 223(2) (2010), 689–727.10.1016/j.aim.2009.09.001CrossRefGoogle Scholar