• DOI:10.1017/S0013091512000338
• Corpus ID: 73722487

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

@article{Hazlewood2011RemarksOS,
  title={Remarks on some fundamental results about higher-rank graphs and their C*-algebras},
  author={Robert Hazlewood and Iain Raeburn and Aidan Sims and Samuel B. G. Webster},
  journal={Proceedings of the Edinburgh Mathematical Society},
  year={2011},
  volume={56},
  pages={575 - 597}
}

• R. Hazlewood, I. Raeburn, +1 author Samuel B. G. Webster
• Published 11 October 2011
• Mathematics
• Proceedings of the Edinburgh Mathematical Society

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 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.