On classifying von Neumann algebras with respect to some property

Question

For which class of von Neumann algebras have the property$:T \in B(\mathcal{H})\text{ such that } \text{Co}_{M}(T)^{-}\cap M'\text{ nonempty }$. Where $\text{Co}_{M}(T)^{-}$ is weak operator closure of convex hull of $\{uTu^{*}:u \in \mathcal{u}(M)\}$, where $M$ is vN algebra in $B(\mathcal{H})$.

Answer 1

It is always nonempty. Even if you take norm closure instead of wot closure. That's exactly what the Dixmier Approximation Theorem says: the set $ \overline{\operatorname{Co}_M}(T), $ where the closure is in norm, always intersects the centre of $M$. So there is always $X\in\overline{\operatorname{Co}_M}(T)$ such that $X\in M\cap M'$.