$C^{\ast }$-algebras associated to coverings of $k$-graphs

Abstract

A covering of $k$-graphs (in the sense of Pask–Quigg–Raeburn) induces an embedding of universal $C^{\ast }$-algebras. We show how to build a $(k+1)$-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of $k$-graph algebras under embeddings induced from coverings as the universal algebra of a $(k+1)$-graph. Our main focus is on computing the $K$-theory of the $(k+1)$-graph algebra from that of the component $k$-graph algebras. Examples of our construction include a realisation of the Kirchberg algebra ${\mathcal{P}}_n$ whose $K$-theory is opposite to that of ${\mathcal{O}}_n$, and a class of A$\mathbf{T}$-algebras that can naturally be regarded as higher-rank Bunce–Deddens algebras.