Orthigonal projections in a von Neumann algebra

Question

Let $M$ be a von Neumann algebra. $p$ is a projection in $M$, it is trivial that $1-p$ is orthogonal to $p$. Does there exist a largest projection $q\in M$ such that $qp=0$?

Answer 1

Yes, it's $1-p$. If $q\geq 1-p$ and $qp=0$, then $$ 1-p=(1-p)q=q-pq=q. $$