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

Journal Article


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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.

UOW Authors


  •   Hazlewood, Robert (external author)
  •   Raeburn, Iain F. (external author)
  •   Sims, Aidan
  •   Webster, Samuel B. (external author)

Publication Date


  • 2013

Citation


  • Hazlewood, R., Raeburn, I., Sims, A. & Webster, S. (2013). Remarks on some fundamental results about higher-rank graphs and their C*-algebras. Proceedings of the Edinburgh Mathematical Society, 56 (2), 575-597.

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2570&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1561

Number Of Pages


  • 22

Start Page


  • 575

End Page


  • 597

Volume


  • 56

Issue


  • 2

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.

UOW Authors


  •   Hazlewood, Robert (external author)
  •   Raeburn, Iain F. (external author)
  •   Sims, Aidan
  •   Webster, Samuel B. (external author)

Publication Date


  • 2013

Citation


  • Hazlewood, R., Raeburn, I., Sims, A. & Webster, S. (2013). Remarks on some fundamental results about higher-rank graphs and their C*-algebras. Proceedings of the Edinburgh Mathematical Society, 56 (2), 575-597.

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2570&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1561

Number Of Pages


  • 22

Start Page


  • 575

End Page


  • 597

Volume


  • 56

Issue


  • 2