Kawahigashi's uniqueness conjecture for almost periodic outer flows on the hyperfinite type II_1 factor

Let $R$) be the hyperfinite type $\mathrm{II}_1$ factor, and let $\alpha: \mathbb{R} \curvearrowright R$ be a flow. Let $\sigma: \mathbb{R} \curvearrowright \mathord{\mathrm{M}}_n(\mathbb{C})$ be a faithful flow on a finite-dimensional factor, and let

$$\sigma^{\infty}=\mathbin{\overline{\bigotimes}}_{\mathbb{N}}\sigma$$

be the corresponding infinite tensor product action on $R=\mathbin{\overline{\bigotimes}}_{\mathbb{N}}\mathord{\mathrm{M}}_n(\mathbb{C})$. The flow has the Rokhlin property when, for every $p\in\mathbb{R}$, there is a unitary eigenvector of eigenvalue $p$ in the appropriate equicontinuous part of the central sequence algebra. It is strictly outer when $R'\cap(R\rtimes_\alpha\mathbb{R})=\mathbb{C}1$.

Kawahigashi's uniqueness conjecture. The following assertions are equivalent: $\alpha$ is cocycle conjugate to $\sigma^{\infty}$; $\alpha$ satisfies the Rokhlin property; $\alpha$ is strictly outer; and $\Gamma(\alpha)=\mathbb{R}$ with $\alpha$ outer.

This conjecture describes the main open classification problem for continuous flows on the hyperfinite type $\mathrm{II}_1$ factor. The source notes that uniqueness up to cocycle conjugacy is known under additional hypotheses, including the Rokhlin property and pointwise fixation of a Cartan subalgebra, but the full equivalence remains open.