Every von Neumann algebra $\mathcal M$ is the dual of a unique Banach space $\mathcal M_*$. The Mackey topology on $\mathcal M$ is the topology of uniform convergence on weakly compact subsets of $\mathcal M_*$. Is it known whether given a von Neumann subalgebra $\mathcal N \subseteq \mathcal M$, the Mackey topology on $\mathcal M$ restricts to the Mackey topology on $\mathcal N$?