Uniqueness theorems for topological higher-rank graph $C^*$-algebras
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- by Jean Renault, Aidan Sims, Dana P. Williams and Trent Yeend
- Proc. Amer. Math. Soc. 146 (2018), 669-684
- DOI: https://doi.org/10.1090/proc/13745
- Published electronically: August 31, 2017
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Abstract:
We consider the boundary-path groupoids of topological higher-rank graphs. We show that all such groupoids are topologically amenable. We deduce that the $C^*$-algebras of topological higher-rank graphs are nuclear and prove versions of the gauge-invariant uniqueness theorem and the Cuntz–Krieger uniqueness theorem. We then provide a necessary and sufficient condition for simplicity of a topological higher-rank graph $C^*$-algebra, and a condition under which it is also purely infinite.References
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Bibliographic Information
- Jean Renault
- Affiliation: Département de Mathématiques, Université d’Orléans, et CNRS (UMR 7349 et FR 2964), BP 6759, 45067 Orléans Cedex 2, France
- MR Author ID: 146950
- Email: Jean.Renault@univ-orleans.fr
- Aidan Sims
- Affiliation: School of Mathematics and Applied Statistics, Austin Keane Building (15), University of Wollongong, NSW 2522, Australia
- MR Author ID: 671497
- Email: asims@uow.edu.au
- Dana P. Williams
- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
- MR Author ID: 200378
- Email: dana.williams@Dartmouth.edu
- Trent Yeend
- Affiliation: School of Mathematical and Physical Sciences, Building V, University of Newcastle, Callaghan NSW 2308, Australia
- MR Author ID: 677884
- Email: Trent.Yeend@ihpa.gov.au
- Received by editor(s): September 9, 2012
- Received by editor(s) in revised form: January 6, 2016, and March 23, 2017
- Published electronically: August 31, 2017
- Additional Notes: This research was supported by the Australian Research Council.
- Communicated by: Ken Ono
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 669-684
- MSC (2010): Primary 46L05
- DOI: https://doi.org/10.1090/proc/13745
- MathSciNet review: 3731700