Type III von Neumann Algebras associated with ${Ø}_{\theta }$

Abstract

Let $\Fth$ be a 2 graph generated by $m$ blue edges and $n$ red edges, and $\omega$ be the distinguished faithful state associated with its graph C*-algebra ${Ø}_{\theta }$. In this paper, we characterize the factorness of the von Neumann algebra ${\pi }_{\omega }({Ø}_{\theta })"$ induced from the GNS representation of $\omega$ under a certain condition. Moreover, when ${\pi }_{\omega }({Ø}_{\theta })"$ is a factor, then it is of type III${}_{m^{-\frac{1}{b}}}$ (or III${}_{n^{-\frac{1}{a}}}$) if $\frac{\ln m}{\ln n}\in\bQ$, where $a,b\in\bN$ with $\gcd (a,b)=1$ satisfy $m^a=n^b$, and of type III${}_1$ if $\frac{\ln m}{\ln n}\not\in\bQ$. In the case of $\theta$ being the identity permutation, our condition turns out to be redundant. On the way to our main results, we also obtain the structure of the fixed point algebra ${Ø}_{\theta }^{\sigma }$ of the modular action $\sigma$ from $\omega$. This could be useful in proving the redundancy of our extra condition.

15 pages

References in corpus (2)

• Dilation Theory for Rank 2 Graph Algebras
• Subalgebras of Graph C*-Algebras

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