Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 11, 2016

Reconstruction of graded groupoids from graded Steinberg algebras

Pere Ara EMAIL logo , Joan Bosa , Roozbeh Hazrat and Aidan Sims
From the journal Forum Mathematicum

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.

MSC 2010: 22A22; 20M18; 16S36

Communicated by Manfred Droste


Award Identifier / Grant number: DP150101598

Funding statement: 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.

Acknowledgements

We are very grateful to the referee, whose helpful comments have significantly improved the exposition of the paper, and have also suggested interesting lines of further enquiry.

References

[1] G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293 (2005), 319–334. 10.1016/j.jalgebra.2005.07.028Search in Google Scholar

[2] G. Abrams and M. Tomforde, Isomorphism and Morita equivalence of graph algebras, Trans. Amer. Math. Soc. 363 (2011), 3733–3767. 10.1090/S0002-9947-2011-05264-5Search in Google Scholar

[3] P. Ara, M. Brustenga and G. Cortiñas, K-theory of Leavitt path algebras, Münster J. Math. 2 (2009), 5–34. Search in 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. 10.1016/j.aim.2013.11.009Search in Google Scholar

[5] P. Ara, M. A. Moreno and E. Pardo, Nonstable K-theory for graph algebras, Algebr. Represent. Theory 10 (2007), 157–178. 10.1007/s10468-006-9044-zSearch in 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. 10.1016/j.jpaa.2016.12.032Search in Google Scholar

[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. 10.1017/etds.2015.52Search in 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. 10.1007/s00233-014-9594-zSearch in Google Scholar

[9] L. O. Clark and A. Sims, Equivalent groupoids have Morita equivalent Steinberg algebras, J. Pure Appl. Algebra 219 (2015), 2062–2075. 10.1016/j.jpaa.2014.07.023Search in Google Scholar

[10] R. Exel, Inverse semigroups and combinatorial C-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), 191–313. 10.1007/s00574-008-0080-7Search in Google Scholar

[11] R. Exel, Partial dynamical systems, Fell bundles and applications, preprint (2015), http://arxiv.org/abs/1511.04565. 10.1090/surv/224Search in Google Scholar

[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. 10.1007/s00233-005-0512-2Search in 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. 10.1016/j.jpaa.2016.05.023Search in Google Scholar

[14] A. Kumjian, On C-diagonals, Canad. J. Math. 38 (1986), 969–1008. 10.4153/CJM-1986-048-0Search in Google Scholar

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

[16] A. Kumjian, D. Pask and I. Raeburn, Cuntz–Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), 161–174. 10.2140/pjm.1998.184.161Search in 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. 10.1006/jfan.1996.3001Search in Google Scholar

[18] M. V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, World Scientific, River Edge, 1998. 10.1142/3645Search in Google Scholar

[19] M. V. Lawson, A noncommutative generalization of Stone duality, J. Aust. Math. Soc. 88 (2010), 385–404. 10.1017/S1446788710000145Search in Google Scholar

[20] M. V. Lawson and D. H. Lenz, Pseudogroups and their étale groupoids, Adv. Math. 244 (2013), 117–170. 10.1016/j.aim.2013.04.022Search in Google Scholar

[21] X. Li, Continuous orbit equivalence rigidity, preprint (2015), http://arxiv.org/abs/1503.01704. 10.1017/etds.2016.98Search in Google Scholar

[22] K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras, Kyoto J. Math. 54 (2014), 863–877. 10.1215/21562261-2801849Search in Google Scholar

[23] A. L. T. Paterson, Groupoids, Inverse Semigroups, and Their Operator Algebras, Birkhäuser, Boston, 1999. 10.1007/978-1-4612-1774-9Search in Google Scholar

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

[25] J. Renault, A Groupoid Approach to C-Algebras, Springer, Berlin, 1980. 10.1007/BFb0091072Search in Google Scholar

[26] J. Renault, Cartan subalgebras in C*-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. 10.33232/BIMS.0061.29.63Search in Google Scholar

[27] B. Steinberg, A groupoid approach to discrete inverse semigroup algebras, Adv. Math. 223 (2010), 689–727. 10.1016/j.aim.2009.09.001Search in Google Scholar

[28] M. Tomforde, Uniqueness theorems and ideal structure for Leavitt path algebras, J. Algebra 318 (2007), 270–299. 10.1016/j.jalgebra.2007.01.031Search in Google Scholar

[29] M. Tomforde, Leavitt path algebras with coefficients in a commutative ring, J. Pure Appl. Algebra 215 (2011), 471–484. 10.1016/j.jpaa.2010.04.031Search in Google Scholar

[30] M. Tomforde, The graph algebra problem page, www.math.uh.edu/~tomforde/GraphAlgebraProblems/GraphAlgebraProblemPage.html. Search in Google Scholar

[31] F. Wehrung, Refinement monoids, equidecomposability types, and Boolean inverse semigroups, preprint (2016), https://hal.archives-ouvertes.fr/hal-01197354. 10.1007/978-3-319-61599-8Search in Google Scholar

Received: 2016-3-15
Revised: 2016-8-30
Published Online: 2016-10-11
Published in Print: 2017-9-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 9.12.2022 from https://www.degruyter.com/document/doi/10.1515/forum-2016-0072/html
Scroll Up Arrow