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Aperiodicity and cofinality for finitely aligned higher-rank graphs†

Published online by Cambridge University Press:  10 May 2010

PETER LEWIN  and
AIDAN SIMS
PETER LEWIN
Affiliation:
School of Mathematics and Applied Statistics, Austin Keane Building (15), University of Wollongong, Wollongong, NSW, Australia. e-mail: [email protected]
AIDAN SIMS
Affiliation:
School of Mathematics and Applied Statistics, Austin Keane Building (15), University of Wollongong, Wollongong, NSW, Australia. e-mail: [email protected]
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Abstract

We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove that C*(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 of C*(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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 333 - 350
Copyright
Copyright © Cambridge Philosophical Society 2010

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