Let $A$ be a von Neumann algebra acting on a Hilbert space $H$. Write $\mathcal{B}(H)$ for the bounded linear operators on $H$. Suppose that $\rho:\mathcal{B}(H) \rightarrow \mathcal{B}(H)$ is an inner automorphism of $\mathcal{B}(H)$, that is $\rho(x) = uxu^{*}$ for some unitary operator $u$ on $H$. Suppose furthermore that $\rho$ leaves $A$ invariant, that is $\rho(A) = A$.

When is $\rho|_{A}:A \rightarrow A$ inner? In other words, when does there exist a unitary $u_{A} \in A$ such that $\rho|_{A}(a) = u_{A} a u_{A}^{*}$, for all $a \in A$.

I am not sure if it matters, but I am interested in the case that $A$ is a factor, and we may furthermore assume that $\rho$ also leaves $A'$ invariant.

Furthermore, we assume that the von Neumann algebra $A$ admits a cyclic and separating vector.