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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
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Abstract

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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
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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.

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