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

TL;DR

This paper provides explicit descriptions and characterizations of higher-rank graphs (k-graphs) and their associated C*-algebras, extending known results and simplifying proofs of key properties.

Contribution

It offers an explicit construction of k-graphs from skeletons and commuting squares, and proves isomorphism conditions and simplicity criteria with direct methods.

Findings

Explicit description of k-graphs from skeletons and squares

Isomorphism of k-graphs characterized by skeletons preserving squares

Direct proof of simplicity conditions for C*-algebras of k-graphs

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 {\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.