Non-tracial free graph von Neumann algebras

Given a finite, directed, connected graph $\Gamma$ equipped with a weighting $\mu$ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional $(\mathcal{M}(\Gamma,\mu),\varphi)$. When the weighting $\mu$ is instead on the vertices of $\Gamma$, the first author showed the isomorphism class of $(\mathcal{M}(\Gamma,\mu),\varphi)$ depends only on the data $(\Gamma,\mu)$ and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of $\mathbb{C}$). Moreover, the free dimension of the interpolated free group factor is easily computed from $\mu$. In this paper, we show for a weighting $\mu$ on the edges of $\Gamma$ that the isomorphism class of $(\mathcal{M}(\Gamma,\mu),\varphi)$ depends only on the data $(\Gamma,\mu)$, and is either as in the vertex weighting case or is a free Araki-Woods factor equipped with a scaling of its free quasi-free state (possibly direct sum copies of $\mathbb{C}$). The latter occurs when the subgroup of $\mathbb{R}^+$ generated by $\mu(e_1)\cdots \mu(e_n)$ for loops $e_1\cdots e_n$ in $\Gamma$ is non-trivial, and in this case the point spectrum of the free quasi-free state will be precisely this subgroup. As an application, we give the isomorphism type of some infinite index subfactors considered previously by Jones and Penneys.