Projection and positive element in C$^*$-algebras

Question

Let ‎$‎‎A$ ‎be ‎a‎ ‎$‎‎C^*$-algebra, ‎$‎‎p\in A$ a ‎‎projection.

‎‎‎ Assume ‎that ‎‎$‎‎a$ ‎is a element in ‎$‎‎ \text{Ball}(A_+)$ ‎such ‎that ‎‎$‎‎a‎\leq p‎$‎.

Q: May I‎ ‎say ‎‎$‎‎ap=pa$? Why?‎ ‎

Answer 1

You have $$ 0\leq(1-p)a(1-p)\leq(1-p)p(1-p)=0. $$ Thus $$0=(1-p)a(1-p)=(a^{1/2}(1-p))^*(a^{1/2}(1-p)),$$ and $a^{1/2}(1-p)=0$, from where $a(1-p)=0$; so $a=ap$. Taking adjoints, $a=pa$. If follows that $a=pap$.