how can i prove this if it is true?

Question

Let Z be a central projection in a von neumann algebra A and Q is a finite projection in A. is it true that ZQ is also finite? if yes how can i prove that? thanks for your help.

Answer 1

Suppose that $ZQ$ is not finite. Then there is a partial isometry $V$ in $A$ with $V^*V = ZQ$ and $VV^*<ZQ$. Let $U = V + (I-Z)Q$. Then $U^*U = Q$ and $UU^* = VV^* + (I-Z)Q < Q$, so that $Q$ is not finite.