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

Allen, Stephen; Pask, David; Sims, Aidan

doi:10.1090/S0002-9939-05-07994-3

A dual graph construction for higher-rank graphs, and $K$-theory for finite 2-graphs
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by Stephen Allen, David Pask and Aidan Sims PDF

Proc. Amer. Math. Soc. 134 (2006), 455-464 Request permission

Abstract:

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

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