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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 $B(\mathcal{H})$.

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  • $\begingroup$ What does the $'$ mean in $M'$? $\endgroup$ – rschwieb Sep 7 '18 at 16:07
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    $\begingroup$ @rschwieb commutant of $M$$=$$M'$ $\endgroup$ – mathlover Sep 7 '18 at 16:12
  • $\begingroup$ @rschwieb: that notation is completely standard in von Neumann algebras. I don't think I have never seen any other notation. $\endgroup$ – Martin Argerami Sep 8 '18 at 2:34
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    $\begingroup$ @MartinArgerami Dear Martin: I suppose that would’ve something natural to say if I were disputing something about it, but if you reread it as someone who is asking a naive question about the most overloaded notation in mathematics, I think you might find it somewhat off-putting. Sometimes people who don’t look like newbies can be newbies in you favorite field, but they don’t want to be bitten either. Regards $\endgroup$ – rschwieb Sep 8 '18 at 20:34
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    $\begingroup$ @rschwieb: I wasn't trying to be aggressive in any way, my apologies if it came out like that. I thought it was important to mention because it is such a basic thing when dealing with von Neumann algebras. $\endgroup$ – Martin Argerami Sep 8 '18 at 21:51
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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 $X\in M\cap M'$.

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