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

Abstract

We show that any almost periodic outer flow $\hat{I}\pm :\mathbb{R}\curvearrowright R$ on the hyperfinite type ${\mathrm{II}}_1$ factor with Connes' spectrum $\hat{I}“(\hat{I}\pm )=\mathbb{R}$ 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.

21 pages. v2: Title changed and minor modifications