Some classification results for generalized q-Gaussian von Neumann algebras

To any trace-preserving action σ : G A of a countable discrete group G on a finite von Neumann algebra A and any orthogonal representation π : G → O(ℓ²_R(G)), we associate the generalized q-Gaussian von Neumann algebra A^π_σ Γq(G, K), where K is a Hilbert space. We then prove that if Gi ^σi (Xi, µi) is a p.m.p. free ergodic rigid action with Gi a non-amenable group having the Haagerup property and πi : Gi → O(ℓ²_R(Gi)) is either trivial or given by conjugation for i = 1, 2, then L^∞(X1) ^π_σ1¹ Γq(G1, K1) ^∼= L^∞(X2) ^π_σ2² Γq(G2, K2) implies that the actions G1 X1, G2 X2 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^∞(X) ^π_σ Γq(Fn, K) for suitable free ergodic rigid p.m.p. actions Fn X.