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Uniqueness theorems for topological higher-rank graph $ C^*$-algebras

Authors: Jean Renault, Aidan Sims, Dana P. Williams and Trent Yeend
Journal: Proc. Amer. Math. Soc. 146 (2018), 669-684
MSC (2010): Primary 46L05
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.

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Additional Information

Jean Renault
Affiliation: Département de Mathématiques, Université d’Orléans, et CNRS (UMR 7349 et FR 2964), BP 6759, 45067 Orléans Cedex 2, France

Aidan Sims
Affiliation: School of Mathematics and Applied Statistics, Austin Keane Building (15), University of Wollongong, NSW 2522, Australia

Dana P. Williams
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551

Trent Yeend
Affiliation: School of Mathematical and Physical Sciences, Building V, University of Newcastle, Callaghan NSW 2308, Australia

Keywords: Topological graph, higher-rank graph, groupoid, amenable groupoid, amenability, graph algebra, Cuntz--Krieger algebra
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
Article copyright: © Copyright 2017 American Mathematical Society

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