inclusion of von Neumann algebras with expectation

Question

Let $N\subset M$ be any inclusion of $\sigma$-finite von Neumann algebras with expectation $E_N： M\rightarrow N$.

If $A\subset N\subset M$ is an abelian von Neumann subalgerba with expectation that is maximal abelian in $M$. Can we find a faithful normal state $\omega$ on $M$ such that $\omega\circ E_N=\omega$ and $A\subset N_{\omega}$, where $N_{\omega}$ is the centralizer of $N$.

Answer 1

If there exists a faithful normal conditional expectation $E_A$, take a faithful normal state $\omega''$ on $A$ and define $\omega'=\omega''\circ E_A$. Then put $\omega=\omega'\circ E_N$.

Now if $x\in A$, $y\in N$, $$ \omega(xy)=\omega(E_A(xy))=\omega(xE_A(y))=\omega(E_A(y)x)=\omega(E_A(yx))=\omega(yx). $$