I saw a note on von Neumann algebras, it mentions that every abelian projection $p$ in a von Neumann algebra is finite.
According to the definitions, we know that $pMp$ is abelian, we need to prove that for any projection $q\in M$ such that $q\leq p$ and $q\sim p$, we have $p=q$.
It is easy to see that $pq=qp=q$, how do I use the condition $p\sim q$ to conclude that $p=q$?
Best Answer
If $p=u^*u$ and $q=uu^*$, then $$u=uu^*u=qu=pqu=pu=pup\in pMp.$$ Since $p$ is abelian, it now follows that $$p=u^*u=uu^*=q.$$