Relative bi-exactness and structural results for graph-wreath product von Neumann algebras

TL;DR

This work advances the understanding of rigidity phenomena in graph-product and graph-wreath product von Neumann algebras by establishing relative bi-exactness results via Ozawa’s $C^{*}$-algebraic framework. It shows that graph products $L H_{\Gamma}$ are bi-exact relative to natural subalgebras, and that graph-wreath products $L(H_{\Gamma}\rtimes G)$ inherit relative bi-exactness under assumptions on the actions and isotropy, leading to (AO)-type control. Leveraging these structural results, the paper proves primeness for a broad class of graph-wreath product $II_1$ factors and derives rigidity conclusions for quotient graphs under stable isomorphism, including isomorphism results when vertex groups have Property $(T)$. The findings thus extend the deformation/rigidity toolkit to graph-structured amalgams and crossed products, producing new examples of prime $II_1$ factors and shedding light on how much graph and group data can be recovered from the resulting von Neumann algebras.

Abstract

We study relative bi-exactness of graph product and graph-wreath product group von Neumann algebras. In particular, we obtain the relative bi-exactness for graph product von Neumann algebras $LH_Γ=\ast_{v,Γ} LH_v$ and graph-wreath product von Neumann algebras $L(H_Γ\rtimes G)=(\ast_{v,Γ} LH)\rtimes G$, assuming that the component groups are exact. We adopt the $C^{\ast}$-algebraic method of Ozawa for the proof. As an application, for a certain class of graph-wreath products, we establish the rigidity result for the quotient graph $G\backslashΓ$ under stable isomorphism. Furthermore, we obtain a new family of prime $\mathrm{II}_1$ factors.

Theorems & Definitions (58)

• Theorem A
• Theorem B
• Remark 1.1
• Theorem C
• Corollary D
• Theorem E
• Theorem 2.1: pop3pop1
• Lemma 2.2: vaes
• Lemma 2.3: vaes
• Lemma 2.4: vaes