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.



Questions about special $C^*$-subalgebras and ideals.
It is true that $B$ is a C$^\*$-subalgebra. But it doesn't have to be unital. Consider for example $A=M\_2(\ell^\infty(\mathbb{N}))$, $I=M\_2(c\_0(\mathbb{N}))$, and
$$
p=\begin{bmatrix}1&0 \\\\ 0&0\end{bmatrix}.
$$
Then $pIp$ is $c\_0(\mathbb{N})$, which is not unital. 


