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Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$.

( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a nonzero finite subprojection)

I want to show that $q$ is semifinite. I have proved it by generalized comparability theorem but I have a simple problem.

Proof:

‎ $0 ‎\neq ‎q_{1}‎‎\leq q ‎\Longrightarrow ‎\exists z \in P(Z(M)): q_{1}z‎\lesssim pz \ , \ (1-z)p‎\lesssim (1-z)q_{1}‎‎‎‎‎‎$‎

Since $p$ is finite, $q_1 z$ is also.

If $0\neq q_1 z$ then $‎0 ‎\neq q_{1}z‎\leq‎ ‎q_{1}‎‎\leq q$ and $q_{1}z$ is finite.

Q: What should I do, If $q_{1}z=0$?

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    $\begingroup$ Your question is very unclear. What is $Z$? Central support? It seems that you want to prove a basic fact, so you should be clear at this point what things you already can assume to be true. For example, if you already know that a von Neumann algebra can be decomposed as a direct sum of finite, semi finite and infinite von Neumann algebras, what you write is very easy (but my guess is that this is what you are ultimately trying to understand). $\endgroup$
    – user75274
    Commented Nov 20, 2015 at 18:07
  • $\begingroup$ $z \in Z(M) \cap P(M)$,central projection and $Z(p)=‎\wedge‎ {s \in P(Z(M)):p ‎\leq s‎} $ I want to prove it completely. $\endgroup$
    – alex v
    Commented Nov 20, 2015 at 18:15

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