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# Periodicity in Rank 2 Graph Algebras

Published:2009-12-01
Printed: Dec 2009
• Kenneth R. Davidson
• Dilian Yang
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## Abstract

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$. The periodic $\mathrm{C}^*$-algebras are characterized, and it is shown that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq \mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$ where $\mathfrak{A}$ is a simple $\mathrm{C}^*$-algebra.
 Keywords: higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectation
 MSC Classifications: 47L55 - Representations of (nonselfadjoint) operator algebras 47L30 - Abstract operator algebras on Hilbert spaces 47L75 - Other nonselfadjoint operator algebras 46L05 - General theory of $C^*$-algebras