Some classification results for generalized -Gaussian von Neumann algebras
Tom 278 / 2024
Streszczenie
To any trace-preserving action \sigma : G \curvearrowright A of a countable discrete group G 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 }^{\pi } \Gamma_q(G,K), where K is a Hilbert space. We then prove that if G_i \curvearrowright ^{\sigma_i} (X_i,\mu_i) is a p.m.p. free ergodic rigid action with G_i a non-amenable group having the Haagerup property and \pi_i:G_i \to \mathcal O(\ell_{\mathbb R }^2(G_i)) is either trivial or given by conjugation for i=1,2, then L^{\infty }(X_1) \rtimes_{\sigma_1}^{\pi_1} \Gamma_q(G_1,K_1) \cong L^{\infty }(X_2) \rtimes_{\sigma_2}^{\pi_2} \Gamma_q(G_2,K_2) implies that the actions G_1 \curvearrowright X_1, G_2 \curvearrowright X_2 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^{\infty }(X) \rtimes_{\sigma }^{\pi } \Gamma_q(\mathbb {F}_n,K) for suitable free ergodic rigid p.m.p. actions \mathbb {F}_n \curvearrowright X.
