Modulus of product of projection and an element

Question

Let $A$ be a von Neumann algebra, $x \in A$ and $p$ be a projection with $|p|\le 1$. Is it true that $|ap|\le |a|$?

In the commutative case, we have $|ap|=|a|\ |p|\le |a|$. But I'm not sure what happens in the non-commutative case?

Answer 1

Projections always satisfy $|p|\leq1$.

This fails in any non-commutative von Neumann algebra. Because you can always do a version of $$ a=\begin{bmatrix}0&0\\0&1\end{bmatrix},\qquad\qquad p=\begin{bmatrix} 1/2&1/2\\1/2&1/2\end{bmatrix} $$