A dual graph construction for higher-rank graphs, and $K$-theory for finite 2-graphs

Abstract

Given a $k$-graph $Λ$ and an element $p$ of $\NN^k$, we define the dual $k$-graph, $pΛ$. We show that when $Λ$ is row-finite and has no sources, the $C^*$-algebras $C^*(Λ)$ and $C^*(pΛ)$ coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the $K$-theory of $C^*(Λ)$ when $Λ$ is finite and strongly connected and satisfies the aperiodicity condition.