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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 higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectation
MSC Classifications: 47L55, 47L30, 47L75, 46L05 show english descriptions Representations of (nonselfadjoint) operator algebras
Abstract operator algebras on Hilbert spaces
Other nonselfadjoint operator algebras
General theory of $C^*$-algebras
47L55 - Representations of (nonselfadjoint) operator algebras
47L30 - Abstract operator algebras on Hilbert spaces
47L75 - Other nonselfadjoint operator algebras
46L05 - General theory of $C^*$-algebras
 

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