On solid ergodicity for Gaussian actions

Rémi Boutonnet (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We investigate Gaussian actions through the study of their crossed-product von Neumann algebra. The motivational result is Chifan and Ioana's ergodic decomposition theorem for Bernoulli actions ([4]) that we generalize to Gaussian actions (Theorem A). We also give general structural results (Theorems 3.4 and 3.8) that allow us to get a more accurate result at the level of von Neumann algebras. More precisely, for a large class of Gaussian actions Γ X, we show that any subfactor N of L ∞ (X) ⋊ Γ containing L ∞ (X) is either hyperfinite or is non-Gamma and prime. At the end of the article, we show a similar result for Bogoliubov actions.

Mots clés: equivalence relations; strong ergodicity; deformation/rigidity

Domaines: Algèbres d'opérateurs [math.OA]; Systèmes dynamiques [math.DS]

Rémi Boutonnet. On solid ergodicity for Gaussian actions. Journal of Functional Analysis, 2012, 263, pp.1040 - 1063. ⟨10.1016/j.jfa.2012.05.011⟩. ⟨hal-01446089⟩