strictly positive in unital $C*$ algebra

Question

Let $A$ be a unital $C*$ algebra. $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Prove that $a\in A_+$ is strictly positive iff $a$ is invertable. I proved one direction : if $a$ is invertable, then $a$ is strictly positive , but I have no idea how to prove the other direction.
Thank you

Answer 1

Since $\overline{aAa}=A$, there is $x\in A$ such that $\|axa-1\|<1/2$. Therefore $axa$ is invertible. This implies that $a$ has left and right inverse and consequently it is invertible.