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

Stephen Allen·David Pask·Aidan Sims
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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.

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Stephen Allen, David Pask, Aidan Sims. 2004-02-08. A dual graph construction for higher-rank graphs, and $K$-theory for finite 2-graphs. https://arxiv.org/abs/math/0402126

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