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 $|g|\leq \phi$ ?