Some classification results for generalized $q$-Gaussian von Neumann algebras
Volume 278 / 2024
Abstract
To any trace-preserving action $\sigma : G \curvearrowright A$ of a countable discrete group $G$ on a finite von Neumann algebra $A$ and any orthogonal representation $\pi :G \to \mathcal O(\ell ^2_{\mathbb R }(G))$, we associate the generalized $q$-Gaussian von Neumann algebra $A \rtimes_{\sigma }^{\pi } \Gamma_q(G,K)$, where $K$ is a Hilbert space. We then prove that if $G_i \curvearrowright ^{\sigma_i} (X_i,\mu_i)$ is a p.m.p. free ergodic rigid action with $G_i$ a non-amenable group having the Haagerup property and $\pi_i:G_i \to \mathcal O(\ell_{\mathbb R }^2(G_i))$ is either trivial or given by conjugation for $i=1,2$, then $L^{\infty }(X_1) \rtimes_{\sigma_1}^{\pi_1} \Gamma_q(G_1,K_1) \cong L^{\infty }(X_2) \rtimes_{\sigma_2}^{\pi_2} \Gamma_q(G_2,K_2)$ implies that the actions $G_1 \curvearrowright X_1$, $G_2 \curvearrowright X_2$ are stably OE. Using results of D. Gaboriau and S. Popa we construct continuously many pairwise non-isomorphic von Neumann algebras of the form $L^{\infty }(X) \rtimes_{\sigma }^{\pi } \Gamma_q(\mathbb {F}_n,K)$ for suitable free ergodic rigid p.m.p. actions $\mathbb {F}_n \curvearrowright X$.
