Bernoulli actions of type III$_0$ with prescribed associated flow
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909091446652928 |
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| author | Berendschot, Tey Vaes, Stefaan |
| author_facet | Berendschot, Tey Vaes, Stefaan |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_02471 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Bernoulli actions of type III$_0$ with prescribed associated flow Berendschot, Tey Vaes, Stefaan Dynamical Systems Operator Algebras 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). |
| title | Bernoulli actions of type III$_0$ with prescribed associated flow |
| topic | Dynamical Systems Operator Algebras |
| url | https://arxiv.org/abs/2106.02471 |