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In the book S. Stratila, L. Zsido "LECTURES ON VON NEUMANN ALGEBRAS" authors prove the following proposition (4.10): If $e,f$ are abelian projections in von Neumann algebra $M$ and $z(e)\leq z(f)$, then $e\precsim f$, where $z(x)$ the central projection of $x\in M$. In the proof the authors by comparison theorem assume that $f\leq e$. Actually I don't understand how we use comparison theorem to assume that $f\leq e$. Thank's for any help.

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By comparison, you have a central projection $p$ with $pe\preceq pf$ and $(1-p)f\prec (1-p)e$. What you want is to show that $1-p=0$. So they work with $(1-p)f$ and $(1-p)e$; they take a subprojection $f_0$ of $(1-p)e$ such that $(1-p)f\sim f_0$ and they consider the proper subprojection $f_0$ of $(1-p)e$. As equivalence of projections and multiplying by central projections preserve abelian projections, they still have that $f_0$ and $(1-p)e$ are abelian.

Then they label $f_0$ and $(1-p)e$ as $f$ and $e$.

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  • $\begingroup$ But further they use the fact $f\leq e$ not the fact $f\prec e$. $\endgroup$ Mar 28, 2019 at 17:15
  • $\begingroup$ Well, yes, but that's irrelevant when dealing with equivalence of projections (at least, with any property that survives equivalence of projections). I have edited that into the answer. $\endgroup$ Mar 28, 2019 at 17:40

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