Type III von Neumann Algebras associated with $\O_\theta$

TL;DR

This paper characterizes when the von Neumann algebra associated with a 2-graph C*-algebra is a type III factor, detailing its type based on the logarithmic ratio of graph parameters and exploring the fixed point algebra structure.

Contribution

It provides a complete characterization of the factor type of the von Neumann algebra from 2-graph C*-algebras under certain conditions, including the case when the permutation is the identity.

Findings

The von Neumann algebra is a type III factor under specified conditions.

Type classification depends on the rationality of the ratio of logarithms of graph parameters.

The structure of the fixed point algebra of the modular action is determined.

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 $\O_\theta$. 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.