标题： 图积冯诺依曼代数的刚性 显示英文标题

标题： Rigidity for graph product von Neumann algebras

摘要： 我们建立了与有限简单图$\Gamma$相关的图乘积冯诺依曼代数$M_\Gamma=*_{v,\Gamma}M_v$和迹冯诺依曼代数族$(M_v)_{v\in\Gamma}$的刚性定理。 我们考虑以下三类顶点代数：扩散型、扩散可约型和II$_1$因子。 在这些三个区域中的每一个中，我们展示了一个大的图类$\Gamma,\Lambda$，其中以下情况成立：任何同构$\theta$在$M_\Gamma$和$N_\Lambda$之间确保存在一个图同构$\alpha:\Gamma\to\Lambda$，并且对于每个顶点$v\in\Gamma$，$\theta(M_v)$和$N_{\alpha(v)}$之间有紧密的关系，从双方的强交织（根据 Popa 的意义），到某些情况下的酉共轭。 我们的结果导致了对图积冯诺依曼代数分类以及计算其对称群的广泛应用。 首先，我们获得了右角Artin群的冯诺依曼代数和ICC群的图积的一般分类定理。 我们还提供了一类具有平凡基本群的II$_1$因子，包括所有在至少具有$5$圈长的图上以及没有度数为$0$或$1$的顶点的II$_1$因子的图积。 最后，我们计算了某些II$_1$因子的图积的外自同构群。 显示英文摘要

摘要： We establish rigidity theorems for graph product von Neumann algebras $M_\Gamma=*_{v,\Gamma}M_v$ associated to finite simple graphs $\Gamma$ and families of tracial von Neumann algebras $(M_v)_{v\in\Gamma}$. We consider the following three broad classes of vertex algebras: diffuse, diffuse amenable, and II$_1$ factors. In each of these three regimes, we exhibit a large class of graphs $\Gamma,\Lambda$ for which the following holds: any isomorphism $\theta$ between $M_\Gamma$ and $N_\Lambda$ ensures the existence of a graph isomorphism $\alpha:\Gamma\to\Lambda$, and tight relations between $\theta(M_v)$ and $N_{\alpha(v)}$ for every vertex $v\in\Gamma$, ranging from strong intertwining in both directions (in the sense of Popa), to unitary conjugacy in some cases. Our results lead to a wide range of applications to the classification of graph product von Neumann algebras and the calculation of their symmetry groups. First, we obtain general classification theorems for von Neumann algebras of right-angled Artin groups and of graph products of ICC groups. We also provide a new family of II$_1$ factors with trivial fundamental group, including all graph products of II$_1$ factors over graphs with girth at least $5$ and no vertices of degree $0$ or $1$. Finally, we compute the outer automorphism group of certain graph products of II$_1$ factors.