Ideal in a $C^*$ algebra 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 $AaA$?

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. 
