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Transactions of the American Mathematical Society

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Simplicity of algebras associated to non-Hausdorff groupoids


Authors: Lisa Orloff Clark, Ruy Exel, Enrique Pardo, Aidan Sims and Charles Starling
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 16S99, 16S10, 22A22, 46L05, 46L55
DOI: https://doi.org/10.1090/tran/7840
Published electronically: June 10, 2019
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Abstract: We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the $ C^{*}$-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that the $ C^{*}$-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in $ \mathbb{Z}_2$ is not simple.


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Additional Information

Lisa Orloff Clark
Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
Email: lisa.clark@vuw.ac.nz

Ruy Exel
Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil
Email: exel@mtm.ufsc.br

Enrique Pardo
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campus de Puerto Real, 11510 Puerto Real (Cádiz), Spain
Email: enrique.pardo@uca.es

Aidan Sims
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia
Email: asims@uow.edu.au

Charles Starling
Affiliation: School of Mathematics and Statistics, Carleton University, 4302 Herzberg Laboratories, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6 Canada
Email: cstar@math.carleton.ca

DOI: https://doi.org/10.1090/tran/7840
Keywords: Groupoid $C^*$-algebra, Steinberg algebra, self-similar graph algebra
Received by editor(s): July 18, 2018
Received by editor(s) in revised form: February 25, 2019
Published electronically: June 10, 2019
Additional Notes: The first-named author was partially supported by a Marsden grant from the Royal Society of New Zealand.
The second-named author was partially supported by CNPq.
The third-named author was partially supported by PAI III grant FQM-298 of the Junta de Andalucía, and by the DGI-MINECO and European Regional Development Fund, jointly, through grants MTM2014-53644-P and MTM2017-83487-P.
The fourth-named author was partially supported by the Australian Research Council grant DP150101595.
The fifth-named author was partially supported by a Carleton University internal research grant.
Dedicated: To Frederick Noel Starling
Article copyright: © Copyright 2019 American Mathematical Society