Zhou operator theory book, Kaplanskys formula

Question

In Zhou's operator theory book, Kaplanskys formula has stated that if $P$ and $Q$ are projection in a von neumann algebra $A$ acting on $H$, then $P\vee Q-Q\sim P-P\wedge Q$. In the proof, it says that if $M$ is the range of $P$ and $N$ the range of $Q$, then we have $\ker P(I-Q)=N \oplus (N^\bot \cap M^\bot)$. I don't understand this equality. Please help me. Thanks.

Answer 1

Any element $x$ in $H$ is of the form $x=y+z$, with $y\in N$, $z\in N^\perp$. That is, $(I-Q)y=0$, $(I-Q)z=z$. Now, if $x\in\ker P(I-Q) $, $$ 0=P(I-Q)x=0+Pz. $$ This happens precisely when $z\in \ker P=M^\perp$. So $y\in N$, $z\in N^\perp\cap M^\perp$.