I was reading the book on operator algebra by Kehe Zhu. The proof of theorem 17.7 (page 107) goes like this :
He first considered the set of all non-empty finite subsets of the set of all projections in a Von Neumann algebra A. Then he ordered this set via set inclusion. He called this set $\Lambda$. Now for each $\alpha=\{P_1,P_2,\dots,P_n\}\in\Lambda$ he used the notation $P_\alpha$ to denote the range projection of the operator $P_1+P_2+\dots+P_n$. Then he wrote $\{P_\alpha\}$ is an increasing net in A. Now my question is how ?
My attempt : I know that for projections P, Q in B(H) (where H is a Hilbert space) the following are equivalent : (i) $P\geq Q$, (ii) $R(Q)\subseteq R(P)$.
I was trying to show that (ii) holds for the net. But I was unable to do so. Because apparently closure of $R(P_1+P_2+\dots,P_n)$ is not a subset of the closure of $R(P_1+P_2+\dots+P_m)$ if $m\geq n$. Even if it is in this case, I am unable to show it. Any help will be appreciated.
Edit : I have done this part by observing that if $m\geq n$ then, showing $\overline{R(P_1+\dots+P_n)}\subseteq\overline{R(P_1+\dots+P_m)}$ is equivalent to showing $N(P_1+\dots+P_m)\subseteq N(P_1+\dots P_n)$ which is easy to show by observing that for any $x\in H$ we have : $$\sum_{i=1}^{k}\,\langle P_ix,x\rangle=\sum_{i=1}^{k}\,\|P_ix\|^2$$ for projections $P_i$.
