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arXiv · 2106.02471

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

Tey Berendschot·Stefaan Vaes
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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).

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Tey Berendschot, Stefaan Vaes. 2021-06-04. Bernoulli actions of type III$_0$ with prescribed associated flow. https://doi.org/10.1017/s1474748022000263

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