Bernoulli crossed products without almost periodic weights

TL;DR

This paper extends the classification of noncommutative Bernoulli crossed products to include cases without almost periodic states, showing they are classified by the acting group and the action itself.

Contribution

It introduces a classification result for Bernoulli crossed products with weakly mixing states, removing the restriction to almost periodic states present in prior work.

Findings

Classification of factors by group and action.

Applicable to a large class of groups and amenable factors.

Includes cases with weakly mixing states, broadening previous results.

Abstract

We prove a classification result for a large class of noncommutative Bernoulli crossed products $(P,\phi)^\Lambda \rtimes \Lambda$ without almost periodic states. Our results improve the classification results from [1], where only Bernoulli crossed products built with almost periodic states could be treated. We show that the family of factors $(P,\phi)^\Lambda \rtimes \Lambda$ with $P$ an amenable factor, $\phi$ a weakly mixing state (i.e. a state for which the modular automorphism group is weakly mixing) and $\Lambda$ belonging to a large class of groups, is classified by the group $\Lambda$ and the action $\Lambda \curvearrowright (P, \phi)^\Lambda$, up to state preserving conjugation of the action. [1] Stefaan Vaes and Peter Verraedt. Classification of type III Bernoulli crossed products. Adv. Math., 281:296-332, 2015.