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 $\mathcal N$?

The article below indicates that the answer was unknown at the time of its publication.

Aarnes, J. F., On the Mackey-Topology for a Von Neumann Algebra, Math. Scand. 22(1968), 87-107 http://www.mscand.dk/article.php?id=1864


This recent paper says that it is still unknown: https://arxiv.org/abs/1411.1890.

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