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Groupoids and $ C^*$-algebras for categories of paths


Author: Jack Spielberg
Journal: Trans. Amer. Math. Soc. 366 (2014), 5771-5819
MSC (2010): Primary 46L05; Secondary 20L05
DOI: https://doi.org/10.1090/S0002-9947-2014-06008-X
Published electronically: June 3, 2014
MathSciNet review: 3256184
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Abstract: In this paper we describe a new method of defining $ C^*$-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the $ C^*$-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.


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Additional Information

Jack Spielberg
Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, Arizona 85287-1804
Email: jack.spielberg@asu.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06008-X
Keywords: Cuntz-Krieger algebra, Toeplitz Cuntz-Krieger algebra, groupoid, aperiodicity
Received by editor(s): January 12, 2012
Received by editor(s) in revised form: February 23, 2012, and October 23, 2012
Published electronically: June 3, 2014
Dedicated: Dedicated to the memory of Bill Arveson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.