Type III von Neumann Algebras associated with $Ø_θ$

Let $\Fth$ be a 2 graph generated by $m$ blue edges and $n$ red edges, and $ω$ be the distinguished faithful state associated with its graph C*-algebra $Ø_θ$. In this paper, we characterize the factorness of the von Neumann algebra $π_ω(Ø_θ)"$ induced from the GNS representation of $ω$ under a certain condition. Moreover, when $π_ω(Ø_θ)"$ 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 $θ$ 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 $Ø_θ^σ$ of the modular action $σ$ from $ω$. This could be useful in proving the redundancy of our extra condition.

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