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Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is not proper, in the sense that there is at least one $x\in M$ for which $$\mathbb{E}_{N}(x)\notin \overline{co}^{w.o.t.}\{u x u^{*}: u\in \mathcal{U}(M)\},$$

where the right hand side here is the weak operator closure of the convex hull of the conjugates of $x$ by unitary elements of $M$. References for such examples would be welcome.

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    $\begingroup$ I think there is no such examples. By results of Hiai etc, a positive element $a$ in a $\mathrm{II}_1$ factor is in the norm- or weak*-closure of the convex hull of the unitary orbit of $x$ iff (the trace distribution function of) $a$ is majorized by $x$. That $E_N(x)$ is majorized by $x$ is proved in [Arveson and Kadison; Diagonals of self-adjoint operators] (it is stated there for masas $N$, but the proof works for general $N$). $\endgroup$
    – Narutaka OZAWA
    Commented Jul 12, 2017 at 1:41
  • $\begingroup$ @Narutaka OZAWA: I was hoping that this was the case! Do you happen to have a reference to Hiai's paper? If not, I will find it. $\endgroup$
    – Jon Bannon
    Commented Jul 12, 2017 at 1:46
  • $\begingroup$ @NARUTAKA OZAWA: I was so happy to see what you wrote, I forgot to thank you for it! Thank you! $\endgroup$
    – Jon Bannon
    Commented Jul 12, 2017 at 2:14
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    $\begingroup$ The result is a $\mathrm{II}_1$ analogue of a very well-known fact in matrix theory and seems to be rediscovered a few times. See for e.g., the recent paper of Dykema--Skoufranis for the reference. arxiv.org/abs/1503.05766 $\endgroup$
    – Narutaka OZAWA
    Commented Jul 12, 2017 at 3:06

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