# Positive elements below projections

Let $a$ be a positive element in $A$, where $A$ is a $C^*$-algebra. Let $p\in A$ a projection and suppose $a\leq p$.

Is it true that $ap=pa$? If yes, shouldn't we have $ap=pa=a$, since $(1-p)a(1-p)=0$?

• What does $a \le p$ mean in a $C^\ast$ algebra? – Robert Lewis Feb 4 '14 at 23:31
• $p-a$ positive i.e. there is $x\in A$ such that $p-a=xx^*$. – Alessandro Vignati Feb 4 '14 at 23:44
• Thanks. Just wiki'ed up as well, but I like your definition better. – Robert Lewis Feb 4 '14 at 23:48

Yes, like you said, $a$ lies in the corner $pAp$, for which $p$ is the unit.