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I have already asked this question and no comment(s) received up to now. I am so curious to get feedback concerning the problem.

Let $M$ be a vn Neumann subalgebra in $B(H)$. Let $f$ and $g$ be normal functionals on $B(H)$ and $M$ respectively. Suppose that $f_{|_{M}}=g$ i.e, the restriction of $f$ to $M$ is just $g$.

Let us define the positive linear functional $\phi:M\to \mathbb{C}$ given by $\phi(x)=|f|(x)$, where $|f|$ is the absolute value of the normal functional $f$.

Q. Can we conclude (in general) that $|g|\leq \phi$ ?

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No, such a property does not hold. For instance, you could take $H = \mathbb{C}^2$ and $M \cong \mathbb{C} \oplus \mathbb{C}$ the subalgebra of diagonal matrices in $B(H)$. Denoting by $E : M_2(\mathbb{C}) = B(H) \rightarrow \mathbb{C} \oplus \mathbb{C}$ the conditional expectation given by restricting a matrix to its diagonal, you are asking whether the operator inequality $|E(A)| \leq E(|A|)$ holds for all matrices $A \in M_2(\mathbb{C})$, where $|B| = (B^*B)^{1/2}$. This fails for instance for $$A = \begin{pmatrix} 1 & 1 \\\ 0 & 0\end{pmatrix} \; .$$

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  • $\begingroup$ It is great thanks a lot Stefaan. $\endgroup$
    – ABB
    Sep 4, 2020 at 12:49
  • $\begingroup$ By any proper property on $M$, could such a property satisfy? $\endgroup$
    – ABB
    Sep 4, 2020 at 12:51
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    $\begingroup$ No, the property only holds if $M = \mathbb{C}1$ or $M = B(H)$. Otherwise, pick projections $p \in M$ and $q$ in the commutant $M'$ such that both $pq$ and $(1-p)(1-q)$ are nonzero. Choose a unit vector $\xi \in H$ in the range of $pq$ and choose a unit vector $\eta$ in the range of $(1-p)(1-q)$. Define $f:B(H)\rightarrow \mathbb{C} : f(x) = \langle x\xi,\xi+\eta\rangle$. Then, $g(x) = \langle x\xi,\xi\rangle$ for all $x \in M$. So, $g$ is positive. Also, $|f|(x) = 2^{-1/2}\langle x(\xi + \eta),(\xi+\eta)\rangle$ for all $x \in B(H)$. So, $|g|(p)=1$, $\phi(p)=2^{-1/2}$, and $|g|\not\leq\phi$. $\endgroup$
    – Stefaan Vaes
    Sep 4, 2020 at 13:46

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