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

Details On some fundamental results about higher-rank graphs and their C*-algebras On some fundamental results about higher-rank graphs and their C*-algebras

2011-10-11

arxiv.org/abs/1110.2269v1

Mathematics

Combinatorics

21 pages, two pictures prepared using TiKZ

Scientific paper

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 to 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 {\Lambda} 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 characterisation, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

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