Let $A$ be a $C^*$-algebra and $I$ be a two side closed (essential) ideal of $A$. Suppose that $p \in A\backslash I$ is a non trivial projection. Let $B=pIp$. My questions are:
(1) Is $B$ a $C^*$-subalgebra of $I$?
(2) If (1) is correct, then, is $B$ unital?
(3) If both (1) and (2) are right, then, what is the "unit" of $B$? is it the projection "p"?
Special case of this may be: $A=M(I)$, the multiplier algebra of $C^*$-algebra $I$. Hope any comments for these.