Some classification results for generalized q-gaussian algebras

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 $\pi :G\to \mathcal{O}({\mathrm{\ell }}_{\mathbb{R}}^2(G))$, 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 $G\curvearrowright A=L^{\mathrm{\infty }}(X)$, $G’ \curvearrowright B = L^{\infty}(Y)$ are p.m.p. free ergodic rigid actions, the commutator subgroups $[G,G]$, $[G^{\mathrm{\prime }},G^{\mathrm{\prime }}]$ are ICC, and $G,G^{\mathrm{\prime }}$ belong to a fairly large class of groups (including all non-amenable groups having the Haagerup property), then $A⋊{\mathrm{\Gamma }}_q(G,K)=B⋊{\mathrm{\Gamma }}_q(G^{\mathrm{\prime }},K^{\mathrm{\prime }})$ 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 $G^{\mathrm{\prime }}$ 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}$_n,K)$,forsuitablefreeergodicrigidp\mathrm{.}m\mathrm{.}p\mathrm{.}actions$$\mathbb{F}$_n \curvearrowright X$.