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Forum Mathematicum

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Volume 29, Issue 5

Issues

Reconstruction of graded groupoids from graded Steinberg algebras

Pere Ara
  • Corresponding author
  • Department of Mathematics, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
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/ Joan Bosa
  • School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, G12 8QW Glasgow,United Kingdom
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/ Roozbeh Hazrat / Aidan Sims
Published Online: 2016-10-11 | DOI: https://doi.org/10.1515/forum-2016-0072

Abstract

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies C*-isomorphism of C*-algebras for graphs E and F in which every cycle has an exit.

Keywords: Steinberg algebra; ample groupoid; Leavitt path algebra

MSC 2010: 22A22; 20M18; 16S36

References

  • [1]

    G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293 (2005), 319–334. Google Scholar

  • [2]

    G. Abrams and M. Tomforde, Isomorphism and Morita equivalence of graph algebras, Trans. Amer. Math. Soc. 363 (2011), 3733–3767. Google Scholar

  • [3]

    P. Ara, M. Brustenga and G. Cortiñas, K-theory of Leavitt path algebras, Münster J. Math. 2 (2009), 5–34. Google Scholar

  • [4]

    P. Ara and R. Exel, Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions, Adv. Math. 252 (2014), 748–804. Web of ScienceGoogle Scholar

  • [5]

    P. Ara, M. A. Moreno and E. Pardo, Nonstable K-theory for graph algebras, Algebr. Represent. Theory 10 (2007), 157–178. Google Scholar

  • [6]

    J. H. Brown, L. O. Clark and A. an Huef, Diagonal-preserving ring *-isomorphisms of Leavitt path algebras, preprint (2015), http://arxiv.org/abs/1510.05309.

  • [7]

    N. Brownlowe, T. M. Carlsen and M. F. Whittaker, Graph algebras and orbit equivalence, Ergodic Theory Dynam. Systems (2015), 10.1017/etds.2015.52. Google Scholar

  • [8]

    L. O. Clark, C. Farthing, A. Sims and M. Tomforde, A groupoid generalisation of Leavitt path algebras, Semigroup Forum 89 (2014), 501–517. Web of ScienceGoogle Scholar

  • [9]

    L. O. Clark and A. Sims, Equivalent groupoids have Morita equivalent Steinberg algebras, J. Pure Appl. Algebra 219 (2015), 2062–2075. Web of ScienceGoogle Scholar

  • [10]

    R. Exel, Inverse semigroups and combinatorial C-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), 191–313. Google Scholar

  • [11]

    R. Exel, Partial dynamical systems, Fell bundles and applications, preprint (2015), http://arxiv.org/abs/1511.04565.

  • [12]

    C. Farthing, P. S. Muhly and T. Yeend, Higher-rank graph C*-algebras: An inverse semigroup and groupoid approach, Semigroup Forum 71 (2005), 159–187. Google Scholar

  • [13]

    R. Johansen and A. P. W. Sørensen, The Cuntz splice does not preserve *-isomorphism of Leavitt path algebras over , J. Pure Appl. Algebra 220 (2016), 3966–3983. Google Scholar

  • [14]

    A. Kumjian, On C-diagonals, Canad. J. Math. 38 (1986), 969–1008. Google Scholar

  • [15]

    A. Kumjian and D. Pask, Higher rank graph C-algebras, New York J. Math. 6 (2000), 1–20. Google Scholar

  • [16]

    A. Kumjian, D. Pask and I. Raeburn, Cuntz–Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), 161–174. Google Scholar

  • [17]

    A. Kumjian, D. Pask, I. Raeburn and J. Renault, Graphs, groupoids, and Cuntz–Krieger algebras, J. Funct. Anal. 144 (1997), 505–541. Google Scholar

  • [18]

    M. V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, World Scientific, River Edge, 1998. Google Scholar

  • [19]

    M. V. Lawson, A noncommutative generalization of Stone duality, J. Aust. Math. Soc. 88 (2010), 385–404. Google Scholar

  • [20]

    M. V. Lawson and D. H. Lenz, Pseudogroups and their étale groupoids, Adv. Math. 244 (2013), 117–170. Google Scholar

  • [21]

    X. Li, Continuous orbit equivalence rigidity, preprint (2015), http://arxiv.org/abs/1503.01704.

  • [22]

    K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras, Kyoto J. Math. 54 (2014), 863–877. Google Scholar

  • [23]

    A. L. T. Paterson, Groupoids, Inverse Semigroups, and Their Operator Algebras, Birkhäuser, Boston, 1999. Google Scholar

  • [24]

    A. L. T. Paterson, Graph inverse semigroups, groupoids and their C*-algebras, J. Operator Theory 48 (2002), 645–662. Google Scholar

  • [25]

    J. Renault, A Groupoid Approach to C-Algebras, Springer, Berlin, 1980. Google Scholar

  • [26]

    J. Renault, Cartan subalgebras in C*-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. Google Scholar

  • [27]

    B. Steinberg, A groupoid approach to discrete inverse semigroup algebras, Adv. Math. 223 (2010), 689–727. Web of ScienceGoogle Scholar

  • [28]

    M. Tomforde, Uniqueness theorems and ideal structure for Leavitt path algebras, J. Algebra 318 (2007), 270–299. Web of ScienceGoogle Scholar

  • [29]

    M. Tomforde, Leavitt path algebras with coefficients in a commutative ring, J. Pure Appl. Algebra 215 (2011), 471–484. Web of ScienceGoogle Scholar

  • [30]

    M. Tomforde, The graph algebra problem page, www.math.uh.edu/~tomforde/GraphAlgebraProblems/GraphAlgebraProblemPage.html.

  • [31]

    F. Wehrung, Refinement monoids, equidecomposability types, and Boolean inverse semigroups, preprint (2016), https://hal.archives-ouvertes.fr/hal-01197354.

About the article


Received: 2016-03-15

Revised: 2016-08-30

Published Online: 2016-10-11

Published in Print: 2017-09-01


Funding Source: Australian Research Council

Award identifier / Grant number: DP150101598

The first and second-named authors were partially supported by the grants DGI MICIIN MTM2011-28992-C02-01 and MINECO MTM2014-53644-P. The second author is supported by the Beatriu de Pinós fellowship (2014 BP-A 00123). This research was supported by the Australian Research Council grant DP150101598.


Citation Information: Forum Mathematicum, Volume 29, Issue 5, Pages 1023–1037, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0072.

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