Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-08T21:36:40.919Z Has data issue: false hasContentIssue false
  • English
  • Français

QUOTIENTS OF ÉTALE GROUPOIDS AND THE ABELIANIZATIONS OF GROUPOID C*-ALGEBRAS

Part of: Selfadjoint operator algebras Topological and differentiable algebraic systems

Published online by Cambridge University Press:  07 April 2020

FUYUTA KOMURA
FUYUTA KOMURA*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama223-8522, Japan e-mail: [email protected] Mathematical Science Team, Center for Advanced Intelligence Project (AIP), RIKEN, Nihonbashi 1-chome Mitsui Building, 15th Floor, 1-4-1 Nihonbashi, Chuo-ku, Tokyo103-0027, Japan
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.

In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by a Cuntz–Krieger uniqueness theorem for a universal groupoid C*-algebra.

Keywords

C*-algebrasétale groupoids

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 22A22: Topological groupoids (including differentiable and Lie groupoids)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 1 , August 2021 , pp. 56 - 75
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
© 2020 Australian Mathematical Publishing Association Inc.

Footnotes

Communicated by L. O. Clark

This work was supported by the RIKEN Junior Research Associate Program.

References

Blackadar, B., ‘Shape theory for C*-algebras’, Math. Scand. 56 (1985), 249–275.Google Scholar
Blackadar, B., Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences (Springer, Berlin, 2006).Google Scholar
Brown, J., Clark, L. O., Farthing, C. and Sims, A., ‘Simplicity of algebras associated to étale groupoids’, Semigroup Forum (2) 88 (2014), 433–452.Google Scholar
Clark, L. O., Exel, R., Pardo, E., Sims, A. and Starling, C., ‘Simplicity of algebras associated to non-Hausdorff groupoids’, Trans. Amer. Math. Soc. (5) 372 (2019), 3669–3712.Google Scholar
Connes, A., ‘A survey of foliations and operator algebras’, Proc. Sympos. Pure Math. 38(1) (1982), 521–628.Google Scholar
Exel, R., ‘Inverse semigroups and combinatorial C*-algebras’, Bull. Brazilian Math. Soc., New Series 39 (2007), 191–313.Google Scholar
Exel, R., ‘Non-Hausdorff étale groupoids’, Proc. Amer. Math. Soc. (3) 139 (2011), 897–907.Google Scholar
Exel, R. and Pardo, E., ‘The tight groupoid of an inverse semigroup’, Semigroup Forum (1) 92 (2016), 274–303.Google Scholar
Paterson, A., Groupoids, Inverse Semigroups, and their Operator Algebras, Progress in Mathematics (Birkhäuser, Boston, 2012).Google Scholar
Renault, J., A Groupoid Approach to C*-Algebras, Lecture Notes in Mathematics (Springer, Berlin, 1980).Google Scholar
Renault, J., ‘Cartan subalgebras in C*-algebras’, Irish Math. Soc. Bull. 61 (2008), 29–63.Google 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 (ed. Perera, F.) (CRM Barcelona, Birkhäuser, 2020).Google Scholar