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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A dual graph construction for higher-rank graphs, and $K$-theory for finite 2-graphs


Authors: Stephen Allen, David Pask and Aidan Sims
Journal: Proc. Amer. Math. Soc. 134 (2006), 455-464
MSC (2000): Primary 46L05
Published electronically: June 29, 2005
MathSciNet review: 2176014
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Abstract: Given a $k$-graph $ \Lambda $ and an element $p$ of $\mathbb{N} ^k$, we define the dual $k$-graph, $p\Lambda $. We show that when $\Lambda $ is row-finite and has no sources, the $C^*$-algebras $C^*(\Lambda )$ and $C^*(p\Lambda )$ coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the $K$-theory of $C^*(\Lambda )$ when $\Lambda $ is finite and strongly connected and satisfies the aperiodicity condition.


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Additional Information

Stephen Allen
Affiliation: Department of Mathematics, University of Newcastle, New South Wales 2308, Australia
Email: stephen.allen@studentmail.newcastle.edu.au

David Pask
Affiliation: Department of Mathematics, University of Newcastle, New South Wales 2308, Australia
Email: david.pask@newcastle.edu.au

Aidan Sims
Affiliation: Department of Mathematics, University of Newcastle, New South Wales 2308, Australia
Email: aidan.sims@newcastle.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07994-3
PII: S 0002-9939(05)07994-3
Keywords: Graphs as categories, graph algebra, $C^*$-algebra, $K$-theory
Received by editor(s): March 22, 2004
Received by editor(s) in revised form: September 20, 2004
Published electronically: June 29, 2005
Additional Notes: This research was supported by the Australian Research Council.
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.