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Transactions of the American Mathematical Society

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The tight groupoid of the inverse semigroups of left cancellative small categories


Authors: Eduard Ortega and Enrique Pardo
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
Published electronically: April 29, 2020
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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.


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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
Email: enrique.pardo@uca.es

DOI: https://doi.org/10.1090/tran/8100
Keywords: Left cancellative small category, inverse semigroup, tight representation, tight groupoid, groupoid $C^*$-algebra
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
Article copyright: © Copyright 2020 American Mathematical Society