# about abelian projection in von Neumann algebras

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.

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 $$e$$.
• But further they use the fact $f\leq e$ not the fact $f\prec e$. Mar 28, 2019 at 17:15