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

Tey Berendschot, Stefaan Vaes

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).

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

Abstract

We prove that many, but not all injective factors arise as crossed products by nonsingular Bernoulli actions of the group . 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 matrices. Finally, we construct Poisson suspension actions with prescribed associated flow for any locally compact second countable group that does not have property (T).

Paper Structure

This paper contains 12 sections, 28 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonsingular group actions
  4. Nonsingular Bernoulli actions
  5. Maharam extension and associated flow
  6. Poisson flows and infinite divisibility: proof of Theorem \ref{['thm.almost-periodic-ITPFI-2']}
  7. Tail boundary flows
  8. Poisson flows
  9. Infinitely divisible flows
  10. Nonsingular Bernoulli shifts: proof of Theorem \ref{['thm.main-structure']}
  11. Nonsingular Poisson suspensions: proof of Proposition \ref{['prop.poisson-suspension']}
  12. Bernoulli actions of amenable groups: proof of Theorem \ref{['thm.main-example']}

Key Result

Theorem 1

Let $\mathbb{Z} \curvearrowright (X,\mu) = \prod_{n \in \mathbb{Z}} (X_0,\mu_n)$ be a nonsingular Bernoulli shift such that $(X,\mu)$ is nonatomic and $\mathbb{Z} \curvearrowright (X,\mu)$ is not dissipative. There exists an essentially unique Borel set $C_0 \subset X_0$ such that $C_0^\mathbb{Z} \s

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Lemma 3.1: Cf. CW88
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 45 more