Nonsingular Gaussian actions: beyond the mixing case

TL;DR

This paper extends the understanding of Gaussian actions of groups beyond the mixing case, establishing ergodicity and type classification for weakly mixing representations and providing new examples of type III$_1$ actions.

Contribution

It introduces new methods to prove ergodicity for weakly mixing representations and classifies the type of Gaussian actions in full generality, including novel examples with non-mixing representations.

Findings

Established ergodicity for Gaussian actions with weakly mixing representations.

Determined the Krieger type of Gaussian actions in full generality.

Constructed examples of type III$_1$ ergodic Gaussian actions with non-mixing representations.

Abstract

Every affine isometric action $\alpha$ of a group $G$ on a real Hilbert space gives rise to a nonsingular action $\hat{\alpha}$ of $G$ on the associated Gaussian probability space. In the recent paper [AIM19], several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation $\pi$ of $G$ is mixing. We develop new methods to prove ergodicity when $\pi$ is only weakly mixing. We determine the type of $\hat{\alpha}$ in full generality. Using Cantor measures, we give examples of type III$_1$ ergodic Gaussian actions of $\mathbb{Z}$ whose underlying representation is non mixing, and even has a Dirichlet measure as spectral type. We also provide very general ergodicity results for Gaussian skew product actions.