Physical Sciences and Mathematics Commons^™

Aperiodicity And Cofinality For Finitely Aligned Higher-Rank Graphs, Peter Lewin, Aidan Sims

Professor Aidan Sims

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; …