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Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Part of: Selfadjoint operator algebras Graph theory

Published online by Cambridge University Press:  30 April 2013

Robert Hazlewood ,
Iain Raeburn ,
Aidan Sims  and
Samuel B. G. Webster
Robert Hazlewood
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia (robbiehazlewood@gmail.com)
Iain Raeburn
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand (iraeburn@maths.otago.ac.nz)
Aidan Sims
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia (asims@uow.edu.au; sbgwebster@gmail.com)
Samuel B. G. Webster
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia (asims@uow.edu.au; sbgwebster@gmail.com)
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Abstract

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Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

Keywords

higher-rank graphC*-algebraCuntz–Krieger algebra

MSC classification

Secondary: 05C20: Directed graphs (digraphs), tournaments 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 575 - 597
Copyright
Copyright © Edinburgh Mathematical Society 2013

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