Periodicity in Rank 2 Graph Algebras

TL;DR

This paper analyzes periodicity conditions in rank 2 graph algebras, characterizing when they are aperiodic or periodic, and explores the structure and simplicity of the associated C*-algebras.

Contribution

It provides a detailed characterization of periodic and aperiodic rank 2 graph algebras and describes their C*-algebra structure, including a new proof of simplicity in the aperiodic case.

Findings

Aperiodic rank 2 graph algebras are characterized.

Periodic C*-algebras are isomorphic to a tensor product involving C(T).

A new proof of simplicity for aperiodic cases is provided.

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 $\ca(\Fth)$. The periodic C*-algebras are characterized, and it is shown that $\ca(\Fth) \simeq \rC(\bT) \otimes \fA$ where $\fA$ is a simple C*-algebra.