Some classification results for generalized q-gaussian algebras

Abstract

To any trace preserving action $\sigma: G \curvearrowright A$ of a countable discrete group 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 $A \rtimes_{\sigma} \Gamma_q^{\pi}(G,K)$, where $K$ is an infinite dimensional separable Hilbert space. Specializing to the cases of $\pi$ being trivial or given by conjugation, we then prove that if $G \curvearrowright A = L^{\infty}(X)$, $G' \curvearrowright B = L^{\infty}(Y)$ are p.m.p. free ergodic rigid actions, the commutator subgroups $[G,G]$, $[G',G']$ are ICC, and $G, G'$ belong to a fairly large class of groups (including all non-amenable groups having the Haagerup property), then $A \rtimes \Gamma_q(G,K) = B \rtimes \Gamma_q(G',K')$ implies that $\mathcal R(G \curvearrowright A)$ is stably isomorphic to $\mathcal R(G' \curvearrowright B)$, where $\mathcal R(G \curvearrowright A), \mathcal R(G' \curvearrowright B)$ are the countable, p.m.p. equivalence relations implemented by the actions of $G$ and $G'$ on $A$ and $B$, respectively. Using results of D. Gaboriau and S. Popa we construct continuously many pair-wise non-isomorphic von Neumann algebras of the form $L^{\infty}(X) \rtimes \Gamma_q(\mathbb{F}_n,K)$, for suitable free ergodic rigid p.m.p. actions $\mathbb{F}_n \curvearrowright X$.