Simplicity of 2-graph algebras associated to Dynamical Systems

TL;DR

This paper provides a combinatorial framework for 2-graph algebras linked to dynamical systems, establishing conditions for simplicity and aperiodicity, and demonstrating that their path spaces have zero entropy.

Contribution

It introduces new, practical criteria for aperiodicity in 2-graphs and connects their path spaces to zero-entropy dynamical systems.

Findings

2-graph $C^*$-algebras are simple and purely infinite under aperiodicity.

Aperiodicity conditions depend only on finite vertex sets.

Path spaces of these 2-graphs have zero entropy.

Abstract

We give a combinatorial description of a family of 2-graphs which subsumes those described by Pask, Raeburn and Weaver. Each 2-graph $\Lambda$ we consider has an associated $C^*$-algebra, denoted $C^*(\Lambda)$, which is simple and purely infinite when $\Lambda$ is aperiodic. We give new, straightforward conditions which ensure that $\Lambda$ is aperiodic. These conditions are highly tractable as we only need to consider the finite set of vertices of $\Lambda$ 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.