It's well known that
a von Neumann algebra is a closed linear span of its projections.
Can we require these projections to be pairwise orthogonal? that is, can we find a set $\mathscr P$ consists of some projections of a von Neumann algebra such that $pq=0\forall p,q\in \mathscr P$ and the von Neumann algebra is closed linear spanned by $\mathscr P$?
Best Answer
No, unless the algebra is commutative, because pairwise orthogonal projections commute, so any members of their linear span commute.