Let $M$ be a von Neumann algebra. Suppose $h$ is a positive self-adjoint operator which is affiliated to the center of $M$ . If $p$ is the spectral projection of $h$， we know that $p$ belongs to the center of $M$.
My question : can we conclude that $hp\in M$?
Not necessarily. Think of $p=1$. If, on the other hand, $p$ is the spectral projection corresponding to a bounded Borel set (that is, $p=1_{K}(h)$ for some Borel $K\subset\mathbb C$) then $hp\in M$, as by the Spectral Theorem it is a limit of linear combinations of projections in $M$.