Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the set of all continuous functions vanishing at infinity
My question is : If $P\in M_{n}(C_{0}(X))$ is a projection, then $PM_{n}(C_{0}(X))P$ is a hereditary subalgebra of $M_{n}(C_{0}(X))$?
Recall a C*-subalgebra $B$ of $A$ is said to be hereditary if for any $a\in A_{+}$ and $b\in B_{+}$ the inequality $a\leq b$ implies that $a\in B$.
If $A$ is a C$^*$-algebra and $p\in A$ is a projection, then $pAp$ is hereditary. For this, assume that $A\subset B(H)$, and suppose that $a\leq b$, with $a\in A$, $b\in pAp$. Then, as $p$ is a projection, $b=pbp$. From $pbp-a\geq0$, we have $$ 0\leq(I-p)(pbp-a)(I-p)=-(I-p)a(I-p). $$ This implies that $(I-p)a(I-p)=0$. Writing $a=c^*c$, we have $[c(I-p)]^*c(I-p)=0$, so $c(I-P)=0$; then $a(I-p)=c^*c(I-p)=0$. Taking adjoint, $(I-p)a=0$. Then $a=pap\in pAp$.
(note that the proof does not assume that $I-p\in A$, so it works even if $A$ is not unital)