Uniqueness of almost periodic outer flows on the hyperfinite type $\mathrm{II}_1$ factor

TL;DR

This paper proves the uniqueness of almost periodic outer flows with full spectrum on the hyperfinite type II_1 factor, establishing their Rokhlin property and related structural properties.

Contribution

It demonstrates that such flows are unique up to cocycle conjugacy and introduces a key cocycle perturbation result for type III amenable equivalence relations.

Findings

Flows with full spectrum satisfy the Rokhlin property.

Such flows are unique up to cocycle conjugacy.

Every almost periodic type III_1 factor has an extremal almost periodic faithful normal state.

Abstract

We show that any almost periodic outer flow $\alpha : \mathbb R \curvearrowright R$ on the hyperfinite type $\mathrm{II}_1$ factor with Connes' spectrum $\Gamma(\alpha) = \mathbb R$ satisfies the Rokhlin property and thus is unique up to cocycle conjugacy. The proof relies on a key cocycle perturbation result for type $\mathrm{III}$ amenable equivalence relations. As a byproduct of our methods, we also show that every almost periodic factor of type $\mathrm{III}_1$ with separable predual has an extremal almost periodic faithful normal state.