Bernoulli actions of amenable groups with weakly mixing Maharam extensions

TL;DR

This paper establishes a criterion for Bernoulli actions of amenable groups to have weakly mixing Maharam extensions and demonstrates that all countable amenable groups can admit stable type III_1 Bernoulli actions, answering a recent open question.

Contribution

It introduces a simple criterion for weakly mixing Maharam extensions in Bernoulli actions and proves the existence of stable type III_1 Bernoulli actions for all countable amenable groups.

Findings

Provided a criterion for weakly mixing Maharam extensions in Bernoulli actions.

Showed that every countable amenable group admits a stable type III_1 Bernoulli action.

Answered a recent open question by Vaes and Wahl.

Abstract

We provide a simple criterion for a non-singular and conservative Bernouilli action to have a weakly mixing Maharam extension. As an application, we show that every countable amenable group admits a stable type III_1 Bernoulli action, answering a recent question by Vaes and Wahl.