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

Authors

Robert Hazlewood, The University Of New South Wales
Iain Raeburn, University of WollongongFollow
Aidan Sims, University of WollongongFollow
Samuel B. G Webster, University of WollongongFollow

RIS ID

79402

Publication Details

Hazlewood, R., Raeburn, I., Sims, A. & Webster, S. B. G. (2013). Remarks on some fundamental results about higher-rank graphs and their C*-algebras. Proceedings of the Edinburgh Mathematical Society, 56 575-597.

Abstract

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. 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.