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Étale Groupoids and Steinberg Algebras a Concise Introduction

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Leavitt Path Algebras and Classical K-Theory

Part of the book series: Indian Statistical Institute Series ((INSIS))

Abstract

We give a concise introduction to (discrete) algebras arising from étale groupoids (aka Steinberg algebras) and describe their close relationship with groupoid \(C^*\)-algebras. Their connection to partial group rings via inverse semigroups is also explored.

Keep fibbing and you’ll end up with the truth!

No truth’s ever been discovered without fourteen fibs along the way, if not one hundred and fourteen, and there’s honour in that.

Dostoyevsky, Crime and Punishment.

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Notes

  1. 1.

    Historically, the term r-discrete was used in place of étale and there are some inconsistencies in the literature surrounding these terms.

References

  1. G. Abrams, P. Ara, M. Siles Molina, Leavitt Path Algebras. Lecture Notes in Mathematics, vol. 2191 (Springer, London, 2017)

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Ara, J. Bosa, R. Hazrat, A. Sims, Reconstruction of graded groupoids from graded Steinberg algebras. Forum Math. 29(5), 1023–1037 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. C.J. Ash, T.E. Hall, Inverse semigroups on graphs. Semigroup Forum 11, 140–145 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Bates, D. Pask, I. Raeburn, W. Szymanśki, The \(C^*\)-algebras of row-finite graphs. N. Y. J. Math. 6(307), 324 (2000)

    MathSciNet  MATH  Google Scholar 

  7. E. Batista, Partial actions: what they are and why we care. Bull. Belg. Math. Soc. Simon Stevin 24(1), 35–71 (2017)

    MathSciNet  MATH  Google Scholar 

  8. S. Berberian, Baer \(*\)-rings. Die Grundlehren der mathematischen Wissenschaften, Band 195 (Springer, New York, 1972)

    Google Scholar 

  9. V.M. Beuter, L. Cordeiro, The dynamics of partial inverse semigroup actions, arXiv:1804.00396

  10. V.M. Beuter, D. Gonçalves, The interplay between Steinberg algebras and partial skew rings, arXiv:1706.00127v1

  11. H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes. Math. Ann. 96, 360–366 (1926)

    Article  MATH  Google Scholar 

  12. R. Brown, Topology and Groupoids. Elements of Modern Topology, 3rd edn. (BookSurge LLC, Charleston, 2006)

    Google Scholar 

  13. R. Brown, From groups to groupoids: a brief survey. Bull. Lond. Math. Soc. 19, 113–134 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Brown, L.O. Clark, C. Farthing, A. Sims, Simplicity of algebras associated to étale groupoids. Semigroup Forum 88(2), 433–452 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Brownlowe, A.P.W. Sørensen, \(L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}} \) does not embed in \(L_{2,\mathbb{Z}}\). J. Algebra 456, 1–22 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Buss, R. Exel, Inverse semigroup expansions and their actions on \(C^*\)-algebras. Ill. J. Math. 56, 1185–1212 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. T.M. Carlsen, J. Rout, Diagonal-preserving graded isomorphisms of Steinberg algebras. Commun. Contemp. Math. 20(6), 1750064 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. L.O. Clark, C. Edie-Michell, A. an Huef, A. Sims, Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras. Trans. Am. Math. Soc. 371(8), 5461–5486. arXiv:1601.07238

    Article  MathSciNet  MATH  Google Scholar 

  19. L.O. Clark, R. Exel, E. Pardo, A. Sims, C. Starling, Simplicity of algebras associated to non-Hausdorff groupoids. Trans. Am. Math. Soc. (2018). arXiv:1806.04362

  20. L.O. Clark, R. Hazrat, S.W. Rigby, Strongly graded groupoids and strongly graded Steinberg algebras. J. Algebra. 530, 34–68 (2019). arXiv:1711.04904

    Article  MathSciNet  MATH  Google Scholar 

  21. L.O. Clark, C. Edie-Michell, Uniqueness theorems for Steinberg algebras. Algebr. Represent. Theory 18(4), 907–916 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. L.O. Clark, R. Exel, E. Pardo, A generalised uniqueness theorem and the graded ideal structure of Steinberg algebras. Forum Math. 30(3), 533–552 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Connes, Noncommutative Geometry (Academic Press, San Diego, 1994)

    MATH  Google Scholar 

  26. M. Crainic, I. Moerdijk, A homology theory for étale groupoids. J. Reine Angew. Math. 521, 25–46 (2000)

    MathSciNet  MATH  Google Scholar 

  27. J. Cuntz, Simple \(C^*\)-algebras generated by isometries. Comm. Math. Phys. 57(2), 173–185 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Dokuchaev, R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans. Am. Math. Soc. 357(5), 1931–1952 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. R. Exel, Partial actions of groups and actions of semigroups. Proc. Am. Math. Soc. 126(12), 3481–3494 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Exel, Inverse semigroups and combinatorial \(C^*\)-algebras. Bull. Braz. Math. Soc. New Ser. 39(2), 191–313 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Exel, F. Vieira, Actions of inverse semigroups arising from partial actions of groups. J. Math. Anal. Appl. 363, 86–96 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Giordano, I.F. Putnam, C.F. Skau, Full groups of Cantor minimal systems. Isr. J. Math. 111, 285–320 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Hazrat, H. Li, Homology of étale groupoids, a graded approach, arXiv:1806.03398 [math.KT]

  34. P.J. Higgins, Categories and groupoids. Repr. Theory Appl. Categ. 7, 1–195 (2005)

    Google Scholar 

  35. I. Kaplansky, Rings of Operators (W. A. Benjamin Inc., New York, 1968)

    MATH  Google Scholar 

  36. A. Kumjian, D. Pask, \(C^{*}\)-algebras of directed graphs and group actions. Ergod. Theory Dyn. Syst. 19, 1503–1519 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Kumjian, D. Pask, Higher rank graph \(C^*\)-algebras. N. Y. J. Math. 6(1), 1–20 (2000)

    MathSciNet  MATH  Google Scholar 

  38. A. Kumjian, D. Pask, I. Raeburn, J. Renault, Graphs, groupoids, and Cuntz–Krieger algebras. J. Funct. Anal. 144(2), 505–541 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. M.V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries (World Scientific Publishing Co. Inc., River Edge, NJ, 1998)

    Google Scholar 

  40. W.G. Leavitt, Modules without invariant basis number. Proc. Am. Math. Soc. 8, 322–328 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  41. H. Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces. Proc. Lond. Math. Soc. 104(3), 27–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. H. Matui, Topological full groups of one-sided shifts of finite type. J. Reine Angew. Math. 705, 35–84 (2015)

    MathSciNet  MATH  Google Scholar 

  43. H. Matui, Étale groupoids arising from products of shifts of finite type. Adv. Math. 303, 502–548 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Z. Mesyan, J. Mitchell, The structure of a graph inverse semigroup. Semigroup Forum 93, 111–130 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Ortega, Homology of the Katsura-Exel-Pardo groupoid (2018). arXiv:1806.09297

  46. E. Pardo, The isomorphism problem for Higman–Thompson groups. J. Algebra 344, 172–183 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  47. A.L.T. Paterson, Groupoids, Inverse Semigroups, and Their Operator Algebras, vol. 170 (Birkhäuser, Basel, 1999)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  49. D. Quillen, Higher Algebraic K-Theory. I. Algebraic K-Theory, I: Higher K-Theories (Proceedings of the Conference on Battelle Memorial Institute, Seattle, WA, 1972). Lecture Notes in Mathematics, vol. 341 (Springer, Berlin, 1973), pp. 85–147

    Google Scholar 

  50. J. Renault, A Groupoid Approach to \(C^{*}\)-Algebras. Lecture Notes in Mathematics, vol. 793 (Springer, Berlin, 1980)

    Google Scholar 

  51. J. Renault, Cuntz-like algebras. in Operator Theoretical Methods, Timioara, 1998 (Theta Foundation, Bucharest, 2000), pp. 371–386

    Google Scholar 

  52. J. Renault, Cartan subalgebras in \(C^*\)-algebras. Ir. Math. Soc. Bull. 61, 29–63 (2008)

    MathSciNet  MATH  Google Scholar 

  53. S.W. Rigby, The groupoid approach to Leavitt path algebras. arXiv:1811.02269

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

    Article  MathSciNet  MATH  Google Scholar 

  55. B. Steinberg, Simplicity, primitivity and semiprimitivity of étale groupoid algebras with applications to inverse semigroup algebras. J. Pure Appl. Algebra 220(3), 1035–1054 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  56. B. Steinberg, Chain conditions on étale groupoid algebras with applications to Leavitt path algebras and inverse semigroup algebras. J. Aust. Math. Soc. 104, 403–411 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  57. B. Steinberg, Prime étale groupoid algebras with applications to inverse semigroup and Leavitt path algebras. J. Pure Appl. Algebra 223(6), 2474–2488 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  58. B. Steinberg, Diagonal-preserving isomorphisms of étale groupoid algebras. J. Algebra 518, 412–439 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  59. S.B.G. Webster, The path space of a directed graph. Proc. Am. Math. Soc. 142, 213–225 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  60. A. Weinstein, Groupoids: unifying internal and external symmetry. A tour through some examples. Not. Am. Math. Soc. 43(7), 744–752 (1996)

    Google Scholar 

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Acknowledgements

The authors would like to acknowledge the grant DP160101481 from the Australian Research Council and Marsden grant VUW1514 from the Royal Society of New Zealand.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Lisa Orloff Clark

  2. Centre for Research in Mathematics, Western Sydney University, Sydney, Australia

    Roozbeh Hazrat

Authors
  1. Lisa Orloff Clark
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  2. Roozbeh Hazrat
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Correspondence to Roozbeh Hazrat .

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Editors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology, Cochin, Kerala, India

    A. A. Ambily

  2. Centre for Research in Mathematics, Western Sydney University, Sydney, NSW, Australia

    Roozbeh Hazrat

  3. Statistics and Mathematics Unit, Indian Statistical Institute, Bengaluru, Karnataka, India

    B. Sury

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Clark, L.O., Hazrat, R. (2020). Étale Groupoids and Steinberg Algebras a Concise Introduction. In: Ambily, A., Hazrat, R., Sury, B. (eds) Leavitt Path Algebras and Classical K-Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-15-1611-5_3

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