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 $|g|\leq \phi$ ?



When a normal functional is restricted to a vn Neumann sub-algebra
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
$$A = \begin{pmatrix} 1 & 1 \\\ 0 & 0\end{pmatrix} \; .$$


