On solid ergodicity for Gaussian actions

Abstract

We investigate Gaussian actions through the study of their crossed-product von Neumann algebra. The motivational result is Chifan and Ioanaʼs ergodic decomposition theorem for Bernoulli actions (Chifan and Ioana, 2010 [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.