# Generalized Comparison between two projection in a von Neumann algebra

In Kadison-Ringrose vol II, the authors proved the Comparison Theorem (6.2.7). They describes it as a "generalization" of the corresponding theorem (6.2.6) that hold in a factor. But I cannot fully grasp the way they used this theorem in the subsequent sections. For example, they used the Comparison Theorem (6.2.7) in the following proposition in the book:

6.4.6(ii). Proposition Let $$E, F$$ are two projection in a von Neumann algebra $$\mathscr{R}$$ and $$E$$ is abelian such that $$C_E \le C_F$$. Then $$E\precsim F$$.

Proof. If $$E\npreceq F$$, then, from the comparison theorem there is some (non-zero) central subprojection $$P$$ of $$C_E$$ such that $$PF\prec PE$$...

I cannot understand the way they use the comparison theorem here in the first line of the proof. How exactly does the existence of the $$P(\le C_E)$$ follow from Comparison theorem?

What I understand is that: since $$E\npreceq F$$, neither $$E\sim F$$ nor $$E\prec F$$. So if we apply Comparison theorem 6.2.7 on these $$E$$ and $$F$$ we can say that (in the notaion of Theorem 6.2.7) $$Q$$ obtained from comparison is not equal to $$I$$...But I cannot see further .... Any explanation regarding the proof 6.4.6 (or Comparison Theorem) will be appreciated. Thanks.

The Comparison Theorem says that you can always find pairwise orthogonal central projections $$P,Q,R$$ such that $$P+Q+R=I$$ and $$PF\prec PE,\qquad QF\sim QE,\qquad RE\prec RF.$$ When you have $$E\not\preceq F$$, you can conclude that $$P\ne0$$, which is what they use. This is because, if $$P=0$$, then $$Q+R=I$$ and you have that $$E\preceq F$$, a contradiction.