Classification of type III Bernoulli crossed products

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 $(P,\varphi {)}^{{\mathbb{F}}_n}⋊{\mathbb{F}}_n$, where ${\mathbb{F}}_n$ is the free group and P is an amenable factor with an almost periodic state $\varphi$. 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.

v2: Minor changes, final version, to appear in Advances in Mathematics