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 \(\pi:G \to \mathcal O(\ell^2_{\mathbb{R}}(G))\), 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 \(\mathbb{F}_n \curvearrowright X\).