Ideal in a $C^*$ algebra

Question

Suppose $A$ is a non-unital $C^*$ algebra, $a\in A$, $I$ is the ideal generated by $a$.

In the unital case, $a=1a1\in AaA$. But in the non-unital case, how to show that $a\in A$, can $a$ be expressed by elements in $AaA$?

Answer 1

I'll assume we are talking (as usual) about closed bilateral ideals. Any C$^*$-algebra has an approximate unit $\{e_j\}$: that is, $0\leq e_j$, $\|e_j\|\leq 1$, and $\lim_j e_ja=\lim_jae_j=a$ for all $a\in A$. Then $$ a=\lim_j e_jae_j\in \overline{AaA}, $$ without even needing sums.