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

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

21 pages, two pictures prepared using TiKZ

References in corpus (1)

• Simplicity of C*-algebras associated to higher-rank graphs

Cited by in corpus (8)

• von Neuman algebras of strongly connected higher-rank graphs
• Topological spaces associated to higher-rank graphs
• Topological realizations and fundamental groups of higher-rank graphs
• Deformations of Fell bundles and twisted graph algebras
• $K$-theory for real $k$-graph $C^{\ast }$-algebras
• Higher-rank graphs and the graded $K$-theory of Kumjian-Pask algebras
• Real and complex K-theory for higher rank graph algebras arising from cube complexes
• Quantum isomorphism of $2$-graphs