a question on hereditary $C^*$- subalgebras Let $A$ be a $C^*$-algebra and $a\in A$ be positive. It is known that $\overline{aAa}$ is the hereditary subalgebra generated by $a$. Now, let $f$ be a continuous function on $[0,\|a\|]$ such that $f(0)=0$ and $f(x)>0$ whenever $x>0$.
My question is whether $\overline{f(a)Af(a)}=\overline{aAa}$? Note that the inclusion $\overline{f(a)Af(a)}\subset\overline{aAa}$ is trivial. So how does one prove the converse?
Thanks for all helps!

What follows is an incomplete solution, but perhaps has some merit. One can apply Proposition 2.5 of this paper here, but there appears to be a problem I describe below.

Lemma: Let $X$ be a compact Hausdorff space, $f,g\in C(X)_+$ be two positive functions such that $\text{supp}(g) \subset \text{supp}(f)$. Then, for any $\epsilon >0, \exists h\in C(X)$ such that $\|g-hfg\| < \epsilon$

Now apply the above lemma to the function $f$ and $g(t) =t$. Let $h$ as in the lemma, and let $b = h(a)f(a)a$. Then for any $x\in A, bxb \in f(a)Af(a)$, and
$$
\|axa - bxb\| \leq \|axa-axb\| + \|axb - bxb\| < \epsilon\|x\|(\|a\| + \|b\|)
$$
Now 
suppose one can control $\|b\|$ in terms of $\|a\|$ and $\|f(a)\|$,
then this would imply $axa \in \overline{f(a)Af(a)}$ proving that $\overline{aAa} \subset \overline{f(a)Af(a)}$

The problem then is to control $\|b\|$, which amounts to controlling $\|h\|$ in the above lemma. Going through the proof, one sees that
$$
\|h\| \leq \frac{1}{\delta}
$$
where $\delta > 0$ is obtained by the continuity of $f$. I am not sure if there is a way to control this quantity. However, for certain functions $f$, it is possible (for instance if $f(t) \geq t$ for all $t\in [0,\|a\|]$).
Hope this helps.
