$C^*$-Algebras Associated to Coverings of $k$-Graphs

A covering of $k$-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal $C^*$-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$\TT$-algebras that can naturally be regarded as higher-rank Bunce-Deddens algebras.

2000 Mathematics Subject Classification: Primary 46L05

Keywords and Phrases: Graph algebra; $k$-graph; covering; $K$-theory; $C^*$-algebra

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