Aperiodicity and cofinality for finitely aligned higher-rank graphs

Abstract

AbstractWe introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove thatC*(Λ) is simple if and only if Λ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of Λ in terms of the ideal structure ofC*(Λ). In an appendix we show how our new cofinality condition simplifies in a number of special cases which have been treated previously in the literature; even in these settings our results are new.