Idea about the projections of the $C^*$-algebra $pAp$, where $p$ is a projection in $A$

Question

Let $A$ be a unital $C^*$-algebra and $p$ be a projection in $A$. Now consider the $C^*$-algebra $pAp$. I want to classify the projections in $pAp.$ I know that, if $q \in A$ is a projection such that $pq=qp$, then $pqp$ is a projection in $pAp$, that is $$(pqp)^*=pq^*p=pqp~~\text{ and }~~ (pqp)(pqp)=pqpqp=pqp.$$ Now I want to know, how the projections in $pAp$ looks like$?$ If we consider $P(A)$ and $P(pAp)$ be the set of all projections in $A$ and $pAp$ respectively, then it is clear that $\{pqp:q \in P(A),~ pq=qp\} \subseteq P(pAp)$. But is the other direction is true, that is, $\{pqp:q \in P(A),~ pq=qp\} = P(pAp)?$ Please help me to see the set $P(pAp)$. Thank you for your help.

Answer 1

Every $x\in pAp$ is of the form $$x=pyp$$ hence satisfies $$px=p^2yp=pyp=pyp^2=xp.$$ Moreover, we have $$pxp=p^2yp^2=pyp=x.$$ Hence every $x\in P(pAp)$ is of the form $x=pqp$ for some $q\in P(A)$ such that $pq=qp:$ take $q:=x$ itself.