Classification of type III Bernoulli crossed products

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 $\varphi$. 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.