Jump to ContentJump to Main Navigation
Show Summary Details
More options …

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


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 30, Issue 3

Issues

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

Lisa Orloff Clark / Ruy Exel
  • Corresponding author
  • Departamento de Matem√°tica, Universidade Federal de Santa Catarina, 88040-970 Florian√≥polis SC, Brazil
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Enrique Pardo
  • Departamento de Matem√°ticas, Facultad de Ciencias, Universidad de C√°diz, Campus de Puerto Real, 11510 Puerto Real (C√°diz), Spain
  • Email
  • 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 ūĚíĘ, and a unital commutative ring R, we consider the Steinberg algebra AR‚ĀĘ(ūĚíĘ). First we prove a uniqueness theorem for this algebra and then, when ūĚíĘ is graded by a cocycle, we study graded ideals in AR‚ĀĘ(ūĚíĘ). 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.

Keywords: Steinberg algebra; graded ideal; self-similar graph algebra; Boolean dynamical system

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

References

  • [1]

    P. Ara, M. A. Moreno and E. Pardo, Nonstable K-Theory for graph algebras, Algebr. Represent. Theory 10 (2007), 157‚Äď178. CrossrefWeb¬†of¬†ScienceGoogle¬†Scholar

  • [2]

    T. Bates and D. Pask, C*-algebras of labelled graphs. II. Simplicity results, Math. Scand. 104 (2009), no. 2, 249‚Äď274. Google¬†Scholar

  • [3]

    T. Bates, D. Pask, I. Raeburn and W. SzymaŇĄski, The C*-algebras of row-finite graphs, New York J. Math. 6 (2000), 307‚Äď324. Google¬†Scholar

  • [4]

    J. Brown, L. O. Clark, C. Farthing and A. Sims, Simplicity of algebras associated to √©tale groupoids, Semigroup Forum 88 (2014), 433‚Äď452. CrossrefWeb¬†of¬†ScienceGoogle¬†Scholar

  • [5]

    J. H. Brown, G. Nagy, S. Reznikoff, A. Sims and D. P. Williams, Cartan subalgebras in C*-algebras of Hausdorff √©tale groupoids, Integral Equations Operator Theory 85 (2016), 109‚Äď126. Google¬†Scholar

  • [6]

    T. M. Carlsen, E. Ortega and E. Pardo, C*-algebras associated to Boolean dynamical systems, J. Math. Anal. Appl. 450 (2017), 727‚Äď768. Web¬†of¬†ScienceGoogle¬†Scholar

  • [7]

    L. O. Clark and C. Edie-Michell, Uniqueness theorems for Steinberg algebras, Algebr. Represent. Theory 18 (2015), 907‚Äď916. CrossrefWeb¬†of¬†ScienceGoogle¬†Scholar

  • [8]

    L. O. Clark, C. Edie-Michell, A. an Huef and A. Sims, Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras, preprint (2016), https://arxiv.org/abs/1601.07238.

  • [9]

    L. O. Clark, C. Farthing, A. Sims and M. Tomforde, A groupoid generalisation of Leavitt path algebras, Semigroup Forum 89 (2014), 501‚Äď517. Web¬†of¬†ScienceCrossrefGoogle¬†Scholar

  • [10]

    L. O. Clark, C. Gil Canto and A. Nasr-Isfahani, The cycline subalgebra of a Kumjian‚ÄďPask algebra, Proc. Amer. Math. Soc. 145 (2017), 1969‚Äď1980. Web¬†of¬†ScienceGoogle¬†Scholar

  • [11]

    L. O. Clark, D. Martín Barquero, C. Martín González and M. Siles Molina, Using Steinberg algebras to study decomposability of Leavitt path algebras, Forum Math. (2016), 10.1515/forum-2016-0062. Web of ScienceGoogle Scholar

  • [12]

    R. Exel, Inverse semigroups and combinatorial C*-algebras, Bull. Braz. Math. Soc. 39 (2008), 191‚Äď313. Web¬†of¬†ScienceGoogle¬†Scholar

  • [13]

    R. Exel, Reconstructing a totally disconnected groupoid from its ample semigroup, Proc. Amer. Math. Soc. 138 (2008), 2991‚Äď3001. Web¬†of¬†ScienceGoogle¬†Scholar

  • [14]

    R. Exel, Non-Hausdorff √©tale groupoids, Proc. Amer. Math. Soc. 139 (2011), 897‚Äď907. CrossrefGoogle¬†Scholar

  • [15]

    R. Exel and E. Pardo, The tight groupoid of an inverse semigroup, Semigroup Forum 92 (2016), 274‚Äď303. CrossrefWeb¬†of¬†ScienceGoogle¬†Scholar

  • [16]

    R. Exel and E. Pardo, Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras, Adv. Math. 306 (2017), 1046‚Äď1129. Web¬†of¬†ScienceGoogle¬†Scholar

  • [17]

    C. Gil Canto and A. Nasr-Isfahani, The maximal commutative subalgebra of a Leavitt path algebra, preprint (2015), https://arxiv.org/abs/1510.03992.

  • [18]

    E. Ortega, Simple Cuntz‚ÄďKrieger Boolean algebras, preprint (2016).

  • [19]

    I. Raeburn, Graph Algebras, CBMS Reg. Conf. Ser. Math. 103, American Mathematical Society, Providence, 2005. Google Scholar

  • [20]

    J. Renault, A Groupoid Approach to C*-Algebra, Lecture Notes in Math. 793, Springer, Berlin, 1980. Google Scholar

  • [21]

    B. Steinberg, A groupoid approach to discrete inverse semigroup algebras, Adv. Math. 223 (2010), 689‚Äď727. CrossrefWeb¬†of¬†ScienceGoogle¬†Scholar

  • [22]

    B. Steinberg, Simplicity, primitivity and semiprimitivity of √©tale groupoid algebras with applications to inverse semigroup algebras, J. Pure Appl. Algebra 220 (2016), 1035‚Äď1054. CrossrefGoogle¬†Scholar

About the article


Received: 2016-09-12

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

Funding Source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

Award identifier / Grant number: 301002/2015-0

Funding Source: Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía

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‚Äď552, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI:¬†https://doi.org/10.1515/forum-2016-0197.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the ‚ÄúCitation Alert‚ÄĚ on the top of this page.

[1]
SIMON W. RIGBY
Journal of the Australian Mathematical Society, 2019, Page 1
[2]
Patrik Nystedt and Johan √Ėinert
Journal of Algebra and Its Applications, 2019, Page 2050165
[4]
Lisa Orloff Clark, Roozbeh Hazrat, and Simon W. Rigby
Journal of Algebra, 2019, Volume 530, Page 34
[5]
Viviane Beuter, Daniel Gon√ßalves, Johan √Ėinert, and Danilo Royer
Forum Mathematicum, 2019, Volume 31, Number 3, Page 543
[6]
Benjamin Steinberg
Journal of Algebra, 2019, Volume 518, Page 412
[7]
Patrik Nystedt, Johan √Ėinert, and H√©ctor Pinedo
Journal of Algebra, 2018, Volume 514, Page 1
[8]
Pere Ara, Roozbeh Hazrat, Huanhuan Li, and Aidan Sims
Algebra & Number Theory, 2018, Volume 12, Number 1, Page 131
[9]
Roozbeh Hazrat and Huanhuan Li
Journal of Pure and Applied Algebra, 2018

Comments (0)

Please log in or register to comment.
Log in