Haagerup–Størmer's conjecture on pointwise inner automorphisms of type III_1 factors

Let $M$ be a type ${\rm III}_1$ factor with separable predual. An automorphism $\Theta\in\operatorname{Aut}(M)$ is pointwise inner if, for every normal state $\psi\in M_*$, there is a unitary $u\in\mathcal U(M)$ such that $\Theta(\psi)=u\psi u^*$. For a faithful normal state $\varphi\in M_*$, write $\sigma_t^\varphi$ for its modular automorphism at $t\in\mathbf R$. Haagerup–Størmer's conjecture. For any automorphism $\Theta\in\operatorname{Aut}(M)$, the following are equivalent: $\Theta$ is pointwise inner, and there exist $u\in\mathcal U(M)$ and $t\in\mathbf R$ such that

$$\Theta=\operatorname{Ad}(u)\circ\sigma_t^\varphi.$$

Here $\varphi\in M_*$ is a faithful normal state. The conjecture classifies pointwise inner automorphisms of type ${\rm III}_1$ factors as compositions of inner and modular automorphisms. The paper proves it for a large class of nonamenable almost periodic type ${\rm III}_1$ factors, including all McDuff factors that tensorially absorb $R_\infty$ and all free Araki–Woods factors, but the general statement remains open.