Classification of type III Bernoulli crossed products

TL;DR

This paper classifies type III Bernoulli crossed product factors arising from free groups and amenable factors with almost periodic states, showing they are determined by the free group's rank and Connes's Sd-invariant.

Contribution

It provides a complete classification of certain type III Bernoulli crossed product factors based on free group rank and Sd-invariant, extending to free product groups and generalized Bernoulli actions.

Findings

Factors are classified by free group rank and Sd-invariant.

Results apply to free product groups and generalized Bernoulli actions.

Complete classification of these type III factors achieved.

Abstract

Crossed products with noncommutative Bernoulli actions were introduced by Connes as the first examples of full factors of type III. This article provides a complete classification of the factors $(P,\phi)^{\mathbb{F}_n} \rtimes \mathbb{F}_n$, where $\mathbb{F}_n$ is the free group and P is an amenable factor with an almost periodic state $\phi$. We show that these factors are completely classified by the rank n of the free group $\mathbb{F}_n$ and Connes's Sd-invariant. We prove similar results for free product groups, as well as for classes of generalized Bernoulli actions.