Nonsingular Gaussian actions: Beyond the mixing case

Abstract

Every affine isometric action α of a group G on a real Hilbert space gives rise to a nonsingular action αˆ of G on the associated Gaussian probability space. In the recent paper [2], several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation π of G is mixing. We develop new methods to prove ergodicity when π is only weakly mixing. We determine the type of αˆ in full generality. Using Cantor measures, we give examples of type III1 ergodic Gaussian actions of 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.