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Periodicity in Rank 2 Graph Algebras |
Canad. J. Math. 61(2009), 1239-1261
Published:2009-12-01
Printed: Dec 2009
Printed: Dec 2009
- Kenneth R. Davidson
- Dilian Yang
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 |