Bernoulli actions of type III$_0$ with prescribed associated flow

TL;DR

This paper characterizes the types and associated flows of Bernoulli actions of the group Z, showing how they relate to injective factors, ergodicity, and flow properties, with applications to constructing actions with specific flows.

Contribution

It provides a general framework linking Bernoulli actions, flow structures, and injective factors, including new constructions for prescribed flows and classifications of associated factors.

Findings

Many injective factors arise from Bernoulli actions of Z.

Associated flows of Bernoulli shifts must be infinitely divisible.

All almost periodic flows can be realized as associated flows of Bernoulli actions.

Abstract

We prove that many, but not all injective factors arise as crossed products by nonsingular Bernoulli actions of the group $\mathbb{Z}$. We obtain this result by proving a completely general result on the ergodicity, type and Krieger's associated flow for Bernoulli shifts with arbitrary base spaces. We prove that the associated flow must satisfy a structural property of infinite divisibility. Conversely, we prove that all almost periodic flows, as well as many other ergodic flows, do arise as associated flow of a weakly mixing Bernoulli action of any infinite amenable group. As a byproduct, we prove that all injective factors with almost periodic flow of weights are infinite tensor products of $2 \times 2$ matrices. Finally, we construct Poisson suspension actions with prescribed associated flow for any locally compact second countable group that does not have property (T).