Simplicity of algebras associated to non-Hausdorff groupoids
HTML articles powered by AMS MathViewer
- by Lisa Orloff Clark, Ruy Exel, Enrique Pardo, Aidan Sims and Charles Starling PDF
- Trans. Amer. Math. Soc. 372 (2019), 3669-3712 Request permission
Abstract:
We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the $C^{*}$-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that the $C^{*}$-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in $\mathbb {Z}_2$ is not simple.References
- Joel Anderson, Extensions, restrictions, and representations of states on $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 249 (1979), no. 2, 303–329. MR 525675, DOI 10.1090/S0002-9947-1979-0525675-1
- Jonathan Brown, Lisa Orloff Clark, Cynthia Farthing, and Aidan Sims, Simplicity of algebras associated to étale groupoids, Semigroup Forum 88 (2014), no. 2, 433–452. MR 3189105, DOI 10.1007/s00233-013-9546-z
- Jonathan H. Brown, Gabriel Nagy, Sarah Reznikoff, Aidan Sims, and Dana P. Williams, Cartan subalgebras in $C^*$-algebras of Hausdorff étale groupoids, Integral Equations Operator Theory 85 (2016), no. 1, 109–126. MR 3503181, DOI 10.1007/s00020-016-2285-2
- Lisa Orloff Clark, Ruy Exel, and Enrique Pardo, A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras, Forum Math. 30 (2018), no. 3, 533–552. MR 3794898, DOI 10.1515/forum-2016-0197
- Lisa Orloff Clark, Cynthia Farthing, Aidan Sims, and Mark Tomforde, A groupoid generalisation of Leavitt path algebras, Semigroup Forum 89 (2014), no. 3, 501–517. MR 3274831, DOI 10.1007/s00233-014-9594-z
- Lisa Orloff Clark and Aidan Sims, Equivalent groupoids have Morita equivalent Steinberg algebras, J. Pure Appl. Algebra 219 (2015), no. 6, 2062–2075. MR 3299719, DOI 10.1016/j.jpaa.2014.07.023
- A. Connes, A survey of foliations and operator algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 521–628. MR 679730
- Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
- R. Exel, Non-Hausdorff étale groupoids, Proc. Amer. Math. Soc. 139 (2011), no. 3, 897–907. MR 2745642, DOI 10.1090/S0002-9939-2010-10477-X
- R. Exel, Reconstructing a totally disconnected groupoid from its ample semigroup, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2991–3001. MR 2644910, DOI 10.1090/S0002-9939-10-10346-3
- Ruy Exel, Inverse semigroups and combinatorial $C^\ast$-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191–313. MR 2419901, DOI 10.1007/s00574-008-0080-7
- Ruy Exel and Enrique Pardo, The tight groupoid of an inverse semigroup, Semigroup Forum 92 (2016), no. 1, 274–303. MR 3448414, DOI 10.1007/s00233-015-9758-5
- Ruy Exel and Enrique Pardo, Self-similar graphs, a unified treatment of Katsura and Nekrashevych $\rm C^*$-algebras, Adv. Math. 306 (2017), 1046–1129. MR 3581326, DOI 10.1016/j.aim.2016.10.030
- Jean-Louis Tu, Non-Hausdorff groupoids, proper actions and $K$-theory, Doc. Math. 9 (2004), 565–597. MR 2117427
- Steven Givant and Paul Halmos, Introduction to Boolean algebras, Undergraduate Texts in Mathematics, Springer, New York, 2009. MR 2466574, DOI 10.1007/978-0-387-68436-9
- R. I. Grigorčuk, On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 53–54 (Russian). MR 565099
- R. I. Grigorchuk, Degrees of growth of finitely generated groups, and the theory of invariant means, Mathematics of the USSR-Izvestiya, 25(2) (1985) 259.
- Takeshi Katsura, A construction of actions on Kirchberg algebras which induce given actions on their $K$-groups, J. Reine Angew. Math. 617 (2008), 27–65. MR 2400990, DOI 10.1515/CRELLE.2008.025
- Mahmood Khoshkam and Georges Skandalis, Regular representation of groupoid $C^*$-algebras and applications to inverse semigroups, J. Reine Angew. Math. 546 (2002), 47–72. MR 1900993, DOI 10.1515/crll.2002.045
- Marcelo Laca, Iain Raeburn, Jacqui Ramagge, and Michael F. Whittaker, Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions, J. Funct. Anal. 266 (2014), no. 11, 6619–6661. MR 3192463, DOI 10.1016/j.jfa.2014.03.003
- Volodymyr Nekrashevych, $C^*$-algebras and self-similar groups, J. Reine Angew. Math. 630 (2009), 59–123. MR 2526786, DOI 10.1515/CRELLE.2009.035
- Volodymyr Nekrashevych, Growth of étale groupoids and simple algebras, Internat. J. Algebra Comput. 26 (2016), no. 2, 375–397. MR 3475064, DOI 10.1142/S0218196716500156
- N. Christopher Phillips, Crossed products of the Cantor set by free minimal actions of $\Bbb Z^d$, Comm. Math. Phys. 256 (2005), no. 1, 1–42. MR 2134336, DOI 10.1007/s00220-004-1171-y
- Iain Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. MR 2135030, DOI 10.1090/cbms/103
- Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266
- Jean Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR 2460017
- Charles Starling, Boundary quotients of $\rm C^*$-algebras of right LCM semigroups, J. Funct. Anal. 268 (2015), no. 11, 3326–3356. MR 3336727, DOI 10.1016/j.jfa.2015.01.001
- Benjamin Steinberg, Simplicity, primitivity and semiprimitivity of étale groupoid algebras with applications to inverse semigroup algebras, J. Pure Appl. Algebra 220 (2016), no. 3, 1035–1054. MR 3414406, DOI 10.1016/j.jpaa.2015.08.006
- Benjamin Steinberg, A groupoid approach to discrete inverse semigroup algebras, Adv. Math. 223 (2010), no. 2, 689–727. MR 2565546, DOI 10.1016/j.aim.2009.09.001
Additional Information
- Lisa Orloff Clark
- Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
- MR Author ID: 624226
- Email: lisa.clark@vuw.ac.nz
- Ruy Exel
- Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil
- MR Author ID: 239607
- Email: exel@mtm.ufsc.br
- Enrique Pardo
- Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campus de Puerto Real, 11510 Puerto Real (Cádiz), Spain
- MR Author ID: 345531
- ORCID: 0000-0002-1909-2895
- Email: enrique.pardo@uca.es
- Aidan Sims
- Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia
- MR Author ID: 671497
- Email: asims@uow.edu.au
- Charles Starling
- Affiliation: School of Mathematics and Statistics, Carleton University, 4302 Herzberg Laboratories, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6 Canada
- MR Author ID: 972128
- Email: cstar@math.carleton.ca
- Received by editor(s): July 18, 2018
- Received by editor(s) in revised form: February 25, 2019
- Published electronically: June 10, 2019
- Additional Notes: The first-named author was partially supported by a Marsden grant 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 grants MTM2014-53644-P and MTM2017-83487-P.
The fourth-named author was partially supported by the Australian Research Council grant DP150101595.
The fifth-named author was partially supported by a Carleton University internal research grant. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3669-3712
- MSC (2010): Primary 16S99, 16S10, 22A22, 46L05, 46L55
- DOI: https://doi.org/10.1090/tran/7840
- MathSciNet review: 3988622
Dedicated: To Frederick Noel Starling