I am trying to understand the proof of the following proposition from the book Kadison-Ringrose vol II
Proposition 6.2.4. If $E$ and $F$ are projections in a von Neumann algebra $\mathcal{R}$ such that $E\preceq F$ and $F \preceq E$, then $E \sim F$.
Proof. Let $V$ and $W$ be partial isometries in $\mathcal{R}$ such that $V^*V=E(=E_0), VV^*=F_1\le F (=F_0), \dots$. Since $V$ maps range of $E$ isometrically onto that of $F_1$, $V$ maps the range of $E_1$ isometrically onto that of a subprojection $F_2$ of $F_1 -$ algebraically, $(VE_1)^*(VE_1)=E_1$ and $(VE_1)(VE_1)^*=F_2$.
- I didn't understand the last line in the proof above.
- I have to verify that: "$V$ maps the range of $E_1$ isometrically onto that of a subprojection $F_2$ of $F_1$" holds if and only if "$(VE_1)^*(VE_1)=E_1$ and $(VE_1)(VE_1)^*=F_2$"
if part: Suppose, $(VE_1)^*(VE_1)=E_1$ and $(VE_1)(VE_1)^*=F_2$ holds. This means $VE_1$ is a partial isometry (in $\mathcal{R}$) with initial space range($E_1$) and final space range($F_2)$ this implies, in particular, $V$ maps range of $E_1$ isometrically onto that of $F_2$.
Only if: If $V$ maps the range of $E_1$ isometrically onto that of a subprojection $F_2$ of $F_1$ then we have: $(VE_1)^*(VE_1)=E_1 V^*VE_1=E_1EE_1=E_1$ this is fine but I cannot prove the 2nd relation! $(VE_1)(VE_1)^*=VE_1V^*=???$; We know $V$ maps ran$E_1$ isometrically onto ran$F_2$, but how do I use this??
Can anyone please explain the proof of the "only if" part . Thank you.
EDIT: I think I got it. First of all assume $V$ maps ran$E_1$ isometrically onto ran$F_2$. From the given conditions on $V$ and $E_1$ (as above) we saw that $VE_1$ is a partial isometry with initial projection $E_1$, i.e., with initial space ran$E_1$. But then $(VE_1)(VE_1)^*$ is the final projection of $VE_1$, i.e., $(VE_1)(VE_1)^*$ is the projection on $\mathcal{H}$ whose range is $VE_1(\mathcal{H})$ which equals $F_2(\mathcal{H})$ and so $(VE_1)(VE_1)^*= F_2$.
It looks to me that you are making this too complicated. You have $$ E_1\leq E,\qquad V^*V=E,\qquad VV^*=F_1. $$ All the authors are saying is that $VE_1$ is a partial isometry with initial projection $E_1$ and final projection a subprojection of $F_1$. That is, they define $F_2=VE_1V^*$, and they automatically have that $$ (VE_1)^*VE_1=E_1V^*VE_1=E_1EE_1=E_1, $$ which already gives you that $VE_1$ is a partial isometry. You also have that $$ F_2=VE_1(VE_1)^*=VE_1V^*\leq VV^*=F_1. $$