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.

中文翻译：

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非迹自由图冯诺依曼代数

给定一个有限的、有向的、连通的图 $\Gamma$ 在它的边上配备了一个加权 $\mu$，我们提供了一个冯诺依曼代数的构造，它配备了一个忠实的、正规的、正线性泛函 $(\mathcal{M} (\Gamma,\mu),\varphi)$。当权重 $\mu$ 在 $\Gamma$ 的顶点上时，第一作者表明 $(\mathcal{M}(\Gamma,\mu),\varphi)$ 的同构类只依赖于数据$(\Gamma,\mu)$ 是一个内插的自由群因子，配备了其独特迹线的缩放（可能是 $\mathbb{C}$ 的直接和副本）。此外，内插自由群因子的自由维数很容易从 $\mu$ 计算出来。在本文中，我们证明了 $\Gamma$ 边上的权重 $\mu$ 的同构类 $(\mathcal{M}(\Gamma,\mu),\varphi)$ 仅依赖于数据$(\Gamma,\mu)$, 并且要么是在顶点权重的情况下，要么是一个自由的 Araki-Woods 因子，配备了其自由准自由状态的缩放（可能是 $\mathbb{C}$ 的直接和副本）。后者发生在由 $\mu(e_1)\cdots \mu(e_n)$ for 循环 $\Gamma$ 中的 $e_1\cdots e_n$ 生成的 $\mathbb{R}^+$ 的子群是非平凡的，在这种情况下，自由准自由态的点谱就是这个子群。作为应用，我们给出了Jones和Penneys之前考虑过的一些无限指数子因子的同构类型。在这种情况下，自由准自由态的点谱就是这个子群。作为应用，我们给出了Jones和Penneys之前考虑过的一些无限指数子因子的同构类型。在这种情况下，自由准自由态的点谱就是这个子群。作为应用，我们给出了Jones和Penneys之前考虑过的一些无限指数子因子的同构类型。