The tight groupoid of the inverse semigroups of left cancellative small categories
HTML articles powered by AMS MathViewer
- by Eduard Ortega and Enrique Pardo PDF
- Trans. Amer. Math. Soc. 373 (2020), 5199-5234 Request permission
Abstract:
We fix a path model for the space of filters of the inverse semigroup $\mathcal {S}_\Lambda$ associated to a left cancellative small category $\Lambda$. Then, we compute its tight groupoid, thus giving a representation of its $C^*$-algebra as a (full) groupoid algebra. Using it, we characterize simplicty for these algebras. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.References
- 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
- Erik Bédos, S. Kaliszewski, John Quigg, and Jack Spielberg, On finitely aligned left cancellative small categories, Zappa-Szép products and Exel-Pardo algebras, Theory Appl. Categ. 33 (2018), Paper No. 42, 1346–1406. MR 3909245
- 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
- Nathan Brownlowe, David Pask, Jacqui Ramagge, David Robertson, and Michael F. Whittaker, Zappa-Szép product groupoids and $C^*$-blends, Semigroup Forum 94 (2017), no. 3, 500–519. MR 3648980, DOI 10.1007/s00233-016-9775-z
- A. P. Donsig, J. Gensler, H. King, D. Milan, and R. Wdowinski, On zigzag maps and the path category of an inverse semigroup, arXiv:1811.04124v1, 2018.
- Allan P. Donsig and David Milan, Joins and covers in inverse semigroups and tight $C^{\ast }$-algebras, Bull. Aust. Math. Soc. 90 (2014), no. 1, 121–133. MR 3227137, DOI 10.1017/S0004972713001111
- 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
- R. Exel, Tight and cover-to-joint representations of semilattices and inverse semigroups, arXiv:1903.02911v1.
- R. Exel and E. Pardo, The tight groupoid of an inverse semigroup, Semigroup Forum 92 (2016), 274–303.
- R. Exel and E. Pardo, Self-similar graphs, a unified treatment of Katsura and Nekrashevych $C^*$-algebras, Adv. Math. 306 (2017), 1046–1129.
- R. Exel, E. Pardo and C. Starling, $C^*$-algebras of self-similar graphs over arbitrary graphs, arXiv:1807.01686v1.
- Mark V. Lawson, Inverse semigroups, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. The theory of partial symmetries. MR 1694900, DOI 10.1142/9789812816689
- Jean N. Renault and Dana P. Williams, Amenability of groupoids arising from partial semigroup actions and topological higher rank graphs, Trans. Amer. Math. Soc. 369 (2017), no. 4, 2255–2283. MR 3592511, DOI 10.1090/tran/6736
- Jack Spielberg, $C^\ast$-algebras for categories of paths associated to the Baumslag-Solitar groups, J. Lond. Math. Soc. (2) 86 (2012), no. 3, 728–754. MR 3000828, DOI 10.1112/jlms/jds025
- Jack Spielberg, Groupoids and $C^*$-algebras for categories of paths, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5771–5819. MR 3256184, DOI 10.1090/S0002-9947-2014-06008-X
- J. Spielberg, Groupoids and $C^*$-algebras for left cancellative small categories, Indiana Univ. Math. J. (to appear), arXiv:1712.07720v1, 2017.
- 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
- 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
Additional Information
- Eduard Ortega
- Affiliation: Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway
- Email: eduardo.ortega@ntnu.no
- 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
- Received by editor(s): June 18, 2019
- Received by editor(s) in revised form: January 6, 2020
- Published electronically: April 29, 2020
- Additional Notes: The second-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 grant MTM2017-83487-P
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 5199-5234
- MSC (2010): Primary 46L05; Secondary 46L80, 46L55, 20L05
- DOI: https://doi.org/10.1090/tran/8100
- MathSciNet review: 4127875