Some classification results for generalized q-gaussian algebras

الأصل · EN

To any trace preserving action σ: G A of a countable discrete group on a finite von Neumann algebra A and any orthogonal representation π:G → O(ℓ²ᵣ(G)), we associate the generalized q-gaussian von Neumann algebra A σ Γqπ(G,K), where K is an infinite dimensional separable Hilbert space. Specializing to the cases of π being trivial or given by conjugation, we then prove that if G A = L∞(X), G' B = L∞(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 Γq(G,K) = B Γq(G',K') implies that R(G A) is stably isomorphic to R(G' B), where R(G A), R(G' 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∞(X) Γq(Fₙ,K), for suitable free ergodic rigid p.m.p. actions Fₙ X.