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Reconstruction of graded groupoids from graded Steinberg algebras

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Veröffentlicht/Copyright: 11. Oktober 2016
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 29 Heft 5

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

Communicated by Manfred Droste

Funding source: Australian Research Council

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.2005.07.028Suche 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/S0002-9947-2011-05264-5Suche in Google Scholar

[3] P. Ara, M. Brustenga and G. Cortiñas, K-theory of Leavitt path algebras, Münster J. Math. 2 (2009), 5–34. Suche 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.2013.11.009Suche 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-zSuche 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.2016.12.032Suche 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.2015.52Suche 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-zSuche 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.2014.07.023Suche in Google Scholar

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

[11] R. Exel, Partial dynamical systems, Fell bundles and applications, preprint (2015), http://arxiv.org/abs/1511.04565. 10.1090/surv/224Suche 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/s00233-005-0512-2Suche in Google Scholar

[13] R. Johansen and A. P. W. Sørensen, The Cuntz splice does not preserve *-isomorphism of Leavitt path algebras over ℤ, J. Pure Appl. Algebra 220 (2016), 3966–3983. 10.1016/j.jpaa.2016.05.023Suche in Google Scholar

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

[15] A. Kumjian and D. Pask, Higher rank graph C∗-algebras, New York J. Math. 6 (2000), 1–20. Suche 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.1998.184.161Suche 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.1996.3001Suche in Google Scholar

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

[19] M. V. Lawson, A noncommutative generalization of Stone duality, J. Aust. Math. Soc. 88 (2010), 385–404. 10.1017/S1446788710000145Suche 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.2013.04.022Suche in Google Scholar

[21] X. Li, Continuous orbit equivalence rigidity, preprint (2015), http://arxiv.org/abs/1503.01704. 10.1017/etds.2016.98Suche 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. 10.1215/21562261-2801849Suche in Google Scholar

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

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

[25] J. Renault, A Groupoid Approach to C∗-Algebras, Springer, Berlin, 1980. 10.1007/BFb0091072Suche in Google Scholar

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

[27] B. Steinberg, A groupoid approach to discrete inverse semigroup algebras, Adv. Math. 223 (2010), 689–727. 10.1016/j.aim.2009.09.001Suche 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.2007.01.031Suche 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.2010.04.031Suche in Google Scholar

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

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

Artikel in diesem Heft

  1. Frontmatter
  2. Weyl asymptotics for Hanoi attractors
  3. Reconstruction of graded groupoids from graded Steinberg algebras
  4. Regularity of weak solutions to linear and quasilinear parabolic systems of non-divergence type with non-smooth in time principal matrix: A(t)-caloric method
  5. Mutations of simple-minded systems in Calabi–Yau categories generated by a spherical object
  6. Some homological properties of category 𝒪. IV
  7. Subgroups of direct products closely approximated by direct sums
  8. Takahasi semigroups
  9. Integration of controlled rough paths via fractional calculus
  10. Qualitative properties of positive solutions of quasilinear equations with Hardy terms
  11. Fourth power moment of coefficients of automorphic L-functions for GL(m)
  12. Reversible homogeneous Finsler metrics with positive flag curvature
https://doi.org/10.1515/forum-2016-0072
Schlagwörter für diesen Artikel
Steinberg algebra; ample groupoid; Leavitt path algebra

Artikel in diesem Heft

  1. Frontmatter
  2. Weyl asymptotics for Hanoi attractors
  3. Reconstruction of graded groupoids from graded Steinberg algebras
  4. Regularity of weak solutions to linear and quasilinear parabolic systems of non-divergence type with non-smooth in time principal matrix: A(t)-caloric method
  5. Mutations of simple-minded systems in Calabi–Yau categories generated by a spherical object
  6. Some homological properties of category 𝒪. IV
  7. Subgroups of direct products closely approximated by direct sums
  8. Takahasi semigroups
  9. Integration of controlled rough paths via fractional calculus
  10. Qualitative properties of positive solutions of quasilinear equations with Hardy terms
  11. Fourth power moment of coefficients of automorphic L-functions for GL(m)
  12. Reversible homogeneous Finsler metrics with positive flag curvature
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