Two ordered relations on projections.

Question

Let $A$ be a vn-algebra. Let $p$ and $q$ be two projections. In the literature, we say $p$ is majorised by $q$ if $pq=p$.

Q. Suppose that $q-p$ is a positive element in $A$ (meaning $q-p=x^*x$ for some $x\in A$). Is necessary $p$ majorised by $q$?

Answer 1

If $q-p=x^*x $, then $$-(1-q)p (1-q)=(1-q)(q-p)(1-q)=[x(1-q)]^*x(1-q)\geq0. $$ it follows that $(1-q)p (1-q)=0$. Thus $$0=(1-q)p (1-q)=[p (1-q)]^*p (1-q), $$ and $p (1-q)=0$.