Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Type III von Neumann Algebras associated with $\O_\theta$   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 $\O_\theta$.
  In this paper, we characterize the factorness of the von Neumann algebra
$\pi_\omega(\O_\theta)"$ induced from the GNS representation of $\omega$ under
a certain condition. Moreover, when $\pi_\omega(\O_\theta)"$ 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 $\O_\theta^\sigma$ of the modular action $\sigma$ from $\omega$.
This could be useful in proving the redundancy of our extra condition.

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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat