# $C_{∗}$-algebras associated to coverings of $k$-graphs

### Alex Kumjian

Department of Mathematics (084) School of Mathematics and University of Nevada Applied Statistics Reno NV 89557-0084 University of Wollongong USA NSW 2522### David Pask

AUSTRALIA### Aidan Sims

School of Mathematics and Applied Statistics University of Wollongong NSW 2522 AUSTRALIA

## Abstract

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 $P_{n}$ whose $K$-theory is opposite to that of $O_{n}$, and a class of A$T$-algebras that can naturally be regarded as higher-rank Bunce–Deddens algebras.

## Cite this article

Alex Kumjian, David Pask, Aidan Sims, $C_{∗}$-algebras associated to coverings of $k$-graphs. Doc. Math. 13 (2008), pp. 161–205

DOI 10.4171/DM/247