Suppose there are positive $a,b,c\in A$ with $a,b,c\leq 1$ and $ab=b$ and $bc=c$, and $\text{rank}(c(t))\geq 2n+1$ at any $t\in X$. I want to prove that there is a nonzero projection $p$ such that $ap=p$(or $\|ap-p\|<\delta$). But I have no knowledge on this. Any help will be apprieciated.
This is one step from the proof of this statement:
$A$ is unital simple and for every finite subset $\mathcal F$ and $\epsilon>0$ there is unital subalgebra $C$ with $1_C=p\in A$ that satisfies the following:
$\begin{align}&C\text{ is a hereditary subalgebra of }C(X)\otimes F \\&\text{ for some finite n-dim CW complex }X \text{ and finite dim algebra }F\\&\|px-xp\|<\epsilon\text{ for }x\in \mathcal F\\&\|pxp-y_x\|<\epsilon \text{ for each }x\in\mathcal F \text{ and some }y_x\in C\end{align}$
Then, every hereditary subalgebra of $A$ contains a nonzero projection.
The proof:
Let $B$ be a hereditary subalgebra.
Take positive $1\geq a,b,c\in B$ such that $ab=b$ and $bc=c$. Take orthogonal $e_1,...,e_{2k+1}$ from $C^*(c)$. Let $\mathcal F$ be sufficiently big so that $\{a,b,c,(a)^{1/2},(b)^{1/2},(c)^{1/2},e_1,...,e_{2n+1}\}\subseteq \mathcal F$.
Next, by simplicity, there are $1=x_{j1}^*e_jx_{j1}+...+x_{jj(n)}^*e_jx_{jj(n)}$. Let $\{x_{ji},x_{ji}^*\}\subseteq\mathcal F$ too.
By assumption and some calculations, for sufficiently small $\epsilon$, there are positive $1\geq f_1,f_2,f_3\in C$ such that $f_1f_2=f_2$ and $f_2f_3=f_3$ and there are mutually orthogonal $d_1,...,d_{2k+1}\leq f_3$, with $\|a^{1/2}pa^{1/2}-f_1\|<\delta$ and $\|p-y_{j1}^*d_jy_{j1}+...+y_{jj(n)}^*d_jy_{jj(n)}\|<1/2$ for some $y_{ji}$.
Now, if there is nonezero $q$ such that $qf_1=q$, then there will be
$\|a^{1/2}pa^{1/2}qa^{1/2}pa^{1/2}-q\|<\delta$
and therefore there is a projection in the hereditary subalgebra generated by $a$.
Then it remains to show the existence of such $q$. Since every $d_j$ has nonezero rank and each two are orthogonal, $f_3$ has rank more than $2n+1$. Thus the question arises.