The following question is from $C^*$-Algebras by Example written by Kenneth R. Davidson. The original question is the Problem I.11.
$\mathit{Definition}:$ Say $\mathcal{W}$ is a $C^*$-subalgebra of a $C^*$-Algebra $\mathcal{U}$ and $A, B \in \mathcal{U}, 0 \leq A \leq B$. We call $\mathcal{W}$ hereditary iff $A \in \mathcal{W}$ whenever $B \in \mathcal{W}$
Given a $C^*$-Algebra $\mathcal{U}$ and a positive element $A$, I am asked to show that $\overline{A\,\mathcal{U}A}$ is THE hereditary $C^*$-subalgebra generated by $A$. I showed that it is hereditary but fail to show it is unique. Could anyone provide me some hints? In general if $W$ is a element from a $C^*$-subalgebra generated by $A$, can we know how $W$ look like?
For the second part of the question, which asks every separable hereditary $C^*$-subalgebra of $\mathcal{U}$ has this form. If we let $\mathcal{W}$ be a separable hereditary $C^*$-subalgebra, I might need to show $\mathcal{W} = E_n\,\mathcal{U}\,E_n$ for some $n \in \mathbb{N}$ where $\{E_k\}_{k \in \mathbb{N}}$ is a increasing sequence of positive elements that forms an approximation identity. I do not know how to show that given a fixed $k \in \mathbb{N}, E_n \in \overline{E_k\,\mathcal{U}\,E_k}\,\forall n \in \mathbb{N}$.
To say that $\overline{A\mathfrak AA}$ is the hereditary $C^*$-subalgebra of $\mathfrak A$ generated by $A$, we need to show two things: That $\overline{A\mathfrak AA}$ is a hereditary $C^*$-subalgebra, and that any hereditary $C^*$-subalgebra $\mathcal W$ of $\mathfrak A$ containing $A$ contains $\overline{A\mathfrak AA}$. You claim to have shown the first part. To show the second can be shown by proving that if $X\in \mathfrak A$, then $AXA\in\mathcal W$. A hint to show this: It suffices to assume $X$ is positive, in which case you can use the identity $AXA\leq \|X\|A^2$.
To show the second part of the question, given your approximate identity $\{E_n\}$, let $A=\sum_{n=1}^\infty E_n/2^n$, and show that $\mathcal W=\overline{A\mathfrak A A}$.