Let $A$ be a von Neumann algebra. Let $p$ be a projection in $A$. Suppose that $e$ is a finite projection. Can we determine all types of vn-algebras in which $p-p\wedge(1-e)$ is a finite projection? 


Rem. It seems that when $A=B(H)$, the range of the projection $p-p\wedge(1-e)$ 
is just $\overline{peH}$ which is clearly finite dimentional subspace, since $e$ is a finite
 projection. 



The range projection of product of projections
By parallelogram rule in "Murray-von Numann equivalency" we have $p-p\wedge(1-e)\sim e-e\wedge(1-p)$. Hence, $p-p\wedge(1-e)$ is always finite, if $e$ is finite.


