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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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1435-5337
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Volume 30, Issue 3

# A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras

Lisa Orloff Clark
/ Ruy Exel
• Corresponding author
• Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Enrique Pardo
• Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campus de Puerto Real, 11510 Puerto Real (Cádiz), Spain
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
Published Online: 2017-08-12 | DOI: https://doi.org/10.1515/forum-2016-0197

## Abstract

Given an ample, Hausdorff groupoid $\mathcal{𝒢}$, and a unital commutative ring R, we consider the Steinberg algebra ${A}_{R}\left(\mathcal{𝒢}\right)$. First we prove a uniqueness theorem for this algebra and then, when $\mathcal{𝒢}$ is graded by a cocycle, we study graded ideals in ${A}_{R}\left(\mathcal{𝒢}\right)$. Applications are given for two classes of ample groupoids, namely those coming from actions of groups on graphs, and also to groupoids defined in terms of Boolean dynamical systems.

MSC 2010: 16S99; 16S10; 22A22; 46L05; 46L55

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## About the article

Revised: 2017-04-04

Published Online: 2017-08-12

Published in Print: 2018-05-01

Funding Source: Royal Society of New Zealand

Award identifier / Grant number: 15-UOO-071

Award identifier / Grant number: 301002/2015-0

Award identifier / Grant number: FQM-298

Funding Source: Dirección General de Investigación Científica y Técnica

Award identifier / Grant number: MTM2014-53644-P

Funding Source: European Regional Development Fund

Award identifier / Grant number: MTM2014-53644-P

The first-named author was partially supported by Marsden grant 15-UOO-071 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 grant MTM2014-53644-P.

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 533–552, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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© 2018 Walter de Gruyter GmbH, Berlin/Boston.

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