On solid ergodicity for Gaussian actions

TL;DR

This paper extends ergodic decomposition results from Bernoulli to Gaussian actions, providing structural insights into their von Neumann algebras and classifying subfactors as hyperfinite or non-Gamma prime.

Contribution

It generalizes ergodic decomposition theorems to Gaussian actions and offers a detailed structural classification of subfactors in their crossed-product von Neumann algebras.

Findings

Subfactors containing L^(X) are either hyperfinite or non-Gamma prime.

Generalization of ergodic decomposition to Gaussian actions.

Extension of results to Bogoliubov actions.

Abstract

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 (Ergodic subequivalence relations induced by a Bernoulli action, {\it Geometric and Functional Analysis}{\bf 20} (2010), 53-67) that we generalize to Gaussian actions. We also give general structural results 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 $\Gamma \curvearrowright X$, we show that any subfactor $N$ of $L^\infty(X) \rtimes \Gamma$ containing $L^\infty(X)$ is either hyperfinite or is non-Gamma and prime. At the end of the article, we generalize this result to Bogoliubov actions.