Projection and positive element in C$^*$-algebras Let ‎$‎‎A$ ‎be ‎a‎ ‎$‎‎C^*$-algebra, ‎$‎‎p\in A$  a ‎‎projection.
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Assume ‎that ‎‎$‎‎a$ ‎is  a element in ‎$‎‎ \text{Ball}(A_+)$ ‎such ‎that  ‎‎$‎‎a‎\leq p‎$‎.

Q: May I‎ ‎say ‎‎$‎‎ap=pa$? Why?‎
  ‎


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 $a=pap$. 
