Simplicity of 2-graph algebras associated to Dynamical Systems

Abstract

We give a combinatorial description of a family of 2-graphs which subsumes those described by Pask, Raeburn and Weaver. Each 2-graph $Λ$ we consider has an associated $C^*$-algebra, denoted $C^*(Λ)$, which is simple and purely infinite when $Λ$ is aperiodic. We give new, straightforward conditions which ensure that $Λ$ is aperiodic. These conditions are highly tractable as we only need to consider the finite set of vertices of $Λ$ in order to identify aperiodicity. In addition, the path space of each 2-graph can be realised as a two-dimensional dynamical system which we show must have zero entropy.

19 pages