Aperiodicity and cofinality for finitely aligned higher-rank graphs

TL;DR

This paper introduces new finite-path-based definitions of aperiodicity and cofinality for finitely aligned higher-rank graphs, establishing their equivalence to simplicity of the associated C*-algebras.

Contribution

It provides novel formulations of aperiodicity and cofinality in terms of finite paths, and characterizes the simplicity of C*() in these terms, improving upon previous approaches.

Findings

C*() is simple iff  is aperiodic and cofinal

New finite-path-based definitions of aperiodicity and cofinality

Simplification of cofinality conditions in special cases

Abstract

We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs \Lambda, and prove that C*(\Lambda) is simple if and only if \Lambda 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 \Lambda in terms of the ideal structure of C*(\Lambda). 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.