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

Abstract

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