Rigidity for graph product von Neumann algebras

Abstract

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.