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On twisted higher-rank graph $ C^*$-algebras


Authors: Alex Kumjian, David Pask and Aidan Sims
Journal: Trans. Amer. Math. Soc. 367 (2015), 5177-5216
MSC (2010): Primary 46L05; Secondary 18G60, 55N10
Published electronically: March 3, 2015
MathSciNet review: 3335414
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Abstract: We define the categorical cohomology of a $ k$-graph $ \Lambda $ and show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted $ k$-graph $ C^*$-algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted $ k$-graph $ C^*$-algebra is isomorphic to a twisted groupoid $ C^*$-algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted $ k$-graph $ C^*$-algebras are nuclear and belong to the bootstrap class.


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Additional Information

Alex Kumjian
Affiliation: Department of Mathematics, University of Nevada, Reno, Nevada 89557-0084
Email: alex@unr.edu

David Pask
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, New South Wales 2522, Australia
Email: dpask@uow.edu.au

Aidan Sims
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, New South Wales 2522, Australia
Email: asims@uow.edu.au

DOI: https://doi.org/10.1090/S0002-9947-2015-06209-6
Keywords: Higher-rank graph, $C^*$-algebra, cohomology, groupoid
Received by editor(s): January 31, 2013
Received by editor(s) in revised form: May 15, 2013
Published electronically: March 3, 2015
Additional Notes: This research was supported by the ARC. Part of the work was completed while the first author was employed at the University of Wollongong on the ARC grants DP0984339 and DP0984360.
Dedicated: Dedicated to Marc A. Rieffel on the occasion of his 75th birthday
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.