In the algebra of matrices $M_n(A)$ over a $C^*$ algebra $A$, consider the corner algebra $PM_n(A)P$ for a Hermitian projection $P\in M_n(A)$. Is there any condition known for $P$ to make $PM_n(A)P$ isomorphic to $A$? If not, where should I look for what is known on the structure of $PM_n(A)P$? (This question is aimed at linking the peoperties of $P$ to $PM_n(A)P$ in general, rather than for the case of specific $C^*$ algebras.)

The background: A Morita context from $A$ to itself can be viewed as a left finitely generated projective $A$-module $L$ whose left module maps are themselves isomorphic to $A$ - thus giving a bimodule structure. If $L$ has associated projection matrix $P$ in $K$-theory, then the left module maps are given by the corner algebra $PM_n(A)P$, thus the question of when this is isomorphic to $A$. (This is the noncommutative analogue of a line bundle in geometry.)