Discrete amenable group actions on von Neumann algebras and invariant nuclear C*-subalgebras

Let $G$ be a countable discrete amenable group, ${\cal M}$ a McDuff factor von Neumann algebra, and $A$ a separable nuclear weakly dense C$^*$-subalgebra of ${\cal M}$. We show that if two centrally free actions of $G$ on ${\cal M}$ differ up to approximately inner automorphisms then they are outer conjugate by an approximately inner automorphism, in the operator norm topology, which makes $A$ invariant. In addition, when $A$ is unital, simple, and with a unique tracial state and $\alpha$ is an automorphism of $A$ we also show that the aperiodicity of $\alpha$ on the von Neumann algebra is equivalent to the weak Rohlin property.