# When a normal functional is restricted to a vn Neumann sub-algebra

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 $$|g|\leq \phi$$ ?

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 $$A = \begin{pmatrix} 1 & 1 \\\ 0 & 0\end{pmatrix} \; .$$
• By any proper property on $M$, could such a property satisfy?
• 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 $|g|\not\leq\phi$. Sep 4, 2020 at 13:46