I'm trying to get understand this basic fact about Von Neumann algebras (VNA). I cannot for the life of me understand what's going on in the Corollary to Proposition 2, Chapter 2 of Dixmier's book on VNAs. I hope someone can provide an independent proof or help me understand the one there.
Definition: Let $\mathcal{A} \subset \mathcal{L}(H)$ be a VNA in the space of bounded linear operators on the Hilbert space $H$. If $E \in \mathcal{A}$ is a projection, we define the reduced VNAs \begin{equation} \begin{split} & \mathcal{A}_E:= \left\lbrace E\circ T|_{\text{Im}(E)} \in \mathcal{L}(\text{Im}(E)): T \in \mathcal{A}\right\rbrace \\ & (\mathcal{A}')_{E} := \left\lbrace E\circ S|_{\text{Im}(E)} \in \mathcal{L}(\text{Im}(E)) : S\in \mathcal{A}'\right\rbrace. \end{split} \end{equation} I understand the proof that both of these are VNAs and $(\mathcal{A}')_E = (\mathcal{A}_E)'$. The Corollary I'm confused about:
Corollary: Let $\mathcal{Z}$ denote the center of $\mathcal{A}$, that is $\mathcal{A} \cap \mathcal{A}'$. Let $E \in \mathcal{A}$ be a projection. Then $\mathcal{Z}_E$ is the center of $\mathcal{A}_E$, that is $ \mathcal{A}_E \cap \mathcal{A}'_E$.
(Note $\mathcal{Z}_E$ is the set of operators $E\circ T \in \mathcal{L}(\text{Im}(E))$ where $T \in \mathcal{Z}$.)
Attempt: It is clear that $\mathcal{Z}_E \subset \mathcal{A}_E \cap \mathcal{A}'_E$. For the reverse containment, start with an element $T \in \mathcal{A}_E\cap \mathcal{A}'_E$ and represent it as $T'_E$ for some $T' \in \mathcal{A}'$. Let $F \in \mathcal{Z}$ be the central support of $E$ So that $E\circ F = E$. This last equality shows that we can assume, without loss of generality that $F\circ T' = T'$. The book says to apply the following proposition to $\mathcal{A}_F$ -
Proposition 2: If $\mathcal{A}$ is a VNA and $E \in \mathcal{A}$ is a projection with central support $F$, then the reduction map $ \mathcal{A}' \mapsto \mathcal{A}'_E $ is an isomorphism if and only if $F =I$.
Note that $E\circ F =E$ implies, since $F$ is a central element, that $\text{Im}(E) \subset \text{Im}(F)$. Thus, I presume we are applying the theorem to $E_F \in \mathcal{A}_F$ and saying something about the map \begin{equation} \mathcal{A}'_F \to (\mathcal{A}'_F)_{E_F} = \mathcal{A}'_E. \end{equation}
Question: How exactly do I use Proposition 2 above to see that $T'_F \in \mathcal{A}_F \cap \mathcal{A}'_F$ as in the book? I guess I'd be equally please if someone can provide an alternate reference for the Corollary in question.
(I suggest you don't use $\circ$ to denote the product in a von Neumann algebra. Though it is technically composition if you think of your algebra as acting on a Hilbert space, it is a notation that neither Dixmier nor any other textbook nor paper in von Neumann algebra uses).
$\def\cZ{\mathcal Z}$ $\def\cA{\mathcal A}$ $\def\abajo{\\[0.3cm]}$ When you consider $T'\in\cA_F$, you are now working in a von Neumann algebra where the central carrier of $E$ is the identity. Then Proposition 2 tells you that $\gamma:S'\longmapsto S'E$ is an isomorphism $\cA'\to\cA'_E$. So, for any $S\in\cA'_F$,
\begin{align} T'S&=\gamma^{-1}\big((T'E)\,(SE)\big)=\gamma^{-1}\big((ET'E)\,(ESE)\big) =\gamma^{-1}\big((ESE)(ET'E)\big)\abajo &=\gamma^{-1}\big((SE)\,(T'E)\big)=ST'. \end{align} Thus $T'\in \cZ(\cA'_F)$. Now, for any $S\in\cA'$, $$ T'S=(T'F)S=T'(FSF)=(FSF)T'=S\,(T'F)=ST'. $$ So $T'\in\cZ(\cA')=\cZ(\cA)$. Then $T=T'E\in \cZ(\cA)_E$.