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Amenability of groupoids arising from partial semigroup actions and topological higher rank graphs


Authors: Jean N. Renault and Dana P. Williams
Journal: Trans. Amer. Math. Soc. 369 (2017), 2255-2283
MSC (2010): Primary 22A22, 46L55, 46L05
DOI: https://doi.org/10.1090/tran/6736
Published electronically: April 8, 2016
MathSciNet review: 3592511
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Abstract: We consider the amenability of groupoids $ G$ equipped with a group valued cocycle $ c:G\to Q$ with amenable kernel $ c^{-1}(e)$. We prove a general result which implies, in particular, that $ G$ is amenable whenever $ Q$ is amenable and if there is countable set $ D\subset G$ such that $ c(G^{u})D=Q$ for all $ u\in G^{(0)}$.

We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our methods yield a nice alternative groupoid approach to these important constructions.


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  • [1] C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L'Enseignement Mathématique [Monographs of L'Enseignement Mathématique], vol. 36, L'Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683
  • [2] William Arveson, An invitation to $ C^*$-algebras, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 39. MR 0512360
  • [3] Victor Arzumanian and Jean Renault, Examples of pseudogroups and their $ C^*$-algebras, Operator algebras and quantum field theory (Rome, 1996) Int. Press, Cambridge, MA, 1997, pp. 93-104. MR 1491110
  • [4] Nathan Brownlowe, Aidan Sims, and Sean T. Vittadello, Co-universal $ C^\ast $-algebras associated to generalised graphs, Israel J. Math. 193 (2013), no. 1, 399-440. MR 3038557, https://doi.org/10.1007/s11856-012-0106-0
  • [5] John Crisp and Marcelo Laca, Boundary quotients and ideals of Toeplitz $ C^*$-algebras of Artin groups, J. Funct. Anal. 242 (2007), no. 1, 127-156. MR 2274018 (2007k:46117)
  • [6] Siegfried Echterhoff and Heath Emerson, Structure and $ K$-theory of crossed products by proper actions, Expo. Math. 29 (2011), no. 3, 300-344. MR 2820377, https://doi.org/10.1016/j.exmath.2011.05.001
  • [7] Siegfried Echterhoff and Dana P. Williams, Structure of crossed products by strictly proper actions on continuous-trace algebras, Trans. Amer. Math. Soc. 366 (2014), no. 7, 3649-3673. MR 3192611, https://doi.org/10.1090/S0002-9947-2014-06263-6
  • [8] R. Exel, Tight representations of semilattices and inverse semigroups, Semigroup Forum 79 (2009), no. 1, 159-182. MR 2534230, https://doi.org/10.1007/s00233-009-9165-x
  • [9] Alex Kumjian and David Pask, Higher rank graph $ C^\ast $-algebras, New York J. Math. 6 (2000), 1-20. MR 1745529
  • [10] Marcelo Laca and Iain Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Funct. Anal. 139 (1996), no. 2, 415-440. MR 1402771, https://doi.org/10.1006/jfan.1996.0091
  • [11] Xin Li, Semigroup $ {\rm C}^*$-algebras and amenability of semigroups, J. Funct. Anal. 262 (2012), no. 10, 4302-4340. MR 2900468
  • [12] Xin Li, Nuclearity of semigroup $ C^*$-algebras and the connection to amenability, Adv. Math. 244 (2013), 626-662. MR 3077884, https://doi.org/10.1016/j.aim.2013.05.016
  • [13] Paul S. Muhly and Dana P. Williams, Renault's equivalence theorem for groupoid crossed products, NYJM Monographs, vol. 3, State University of New York University at Albany, Albany, NY, 2008. Available at http://nyjm.albany.edu:8000/m/2008/3.htm.
  • [14] A. Nica, $ C^*$-algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory 27 (1992), no. 1, 17-52. MR 1241114
  • [15] John C. Quigg, Discrete $ C^*$-coactions and $ C^*$-algebraic bundles, J. Austral. Math. Soc. Ser. A 60 (1996), no. 2, 204-221. MR 1375586
  • [16] Arlan Ramsay, The Mackey-Glimm dichotomy for foliations and other Polish groupoids, J. Funct. Anal. 94 (1990), no. 2, 358-374. MR 1081649 (93a:46124)
  • [17] Jean Renault, A groupoid approach to $ C^*$-algebras, Lecture Notes in Mathematics, vol. 793, Springer-Verlag, New York, 1980.
  • [18] Jean Renault, Cuntz-like algebras, Operator theoretical methods (Timişoara, 1998) Theta Found., Bucharest, 2000, pp. 371-386. MR 1770333
  • [19] Jean Renault, Topological amenability is a Borel property, Math. Scand. 117 (2015), no. 1, 5-30. MR 3403785
  • [20] Jean Renault, Aidan Sims, Dana P. Williams, and Trent Yeend, Uniqueness theorems for topological higher-rank graph $ C^*$-algebras, Proc. Amer. Math. Soc., submitted.
  • [21] Aidan Sims and Dana P. Williams, The primitive ideals of some étale groupoid $ C^*$-algebras, Algebras and Representation Theory , posted on (26 September 2015)., https://doi.org/10.1007/s10468-015-9573-4
  • [22] Jack Spielberg, Groupoids and $ C^*$-algebras for categories of paths, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5771-5819. MR 3256184, https://doi.org/10.1090/S0002-9947-2014-06008-X
  • [23] S. Sundar, $ C^*$-algebras associated to Ore semigroups, 2015. (arXiv.math.OA.1408.4242).
  • [24] Jean-Louis Tu, La conjecture de Baum-Connes pour les feuilletages moyennables, $ K$-Theory 17 (1999), no. 3, 215-264. MR 1703305 (2000g:19004)
  • [25] Dana P. Williams, Crossed products of $ C{\sp \ast }$-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954 (2007m:46003)
  • [26] Trent Yeend, Groupoid models for the $ C^*$-algebras of topological higher-rank graphs, J. Operator Theory 57 (2007), no. 1, 95-120. MR 2301938

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Additional Information

Jean N. Renault
Affiliation: Départment de Mathématiques, Université d’Orléans et CNRS (UMR 7349 et FR 2964), 45067 Orléans Cedex 2, France
Email: jean.renault@univ-orleans.fr

Dana P. Williams
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
Email: dana.williams@Dartmouth.edu

DOI: https://doi.org/10.1090/tran/6736
Keywords: Groupoids, amenable groupoids, cocycle, semigroup actions, higher rank graphs, topological higher rank graphs.
Received by editor(s): January 12, 2015
Received by editor(s) in revised form: March 17, 2015
Published electronically: April 8, 2016
Additional Notes: The second author was supported by a grant from the Simons Foundation.
Article copyright: © Copyright 2016 American Mathematical Society

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