Constructing a proper dense subalgebra from a subalgebra inside an ideal

Question

Given a C* algebra $A$ and *-sub-algebra $B$, where $B$ is contained in a two-sided, closed ideal $I$, can we construct a proper, dense sub-algebra that contain $B$?

I am considering the following construction: Given $A$, we can always find a proper dense set $D$($A$ is impossible to equip with discrete topology), then we adjoin $D$ to $B$,so $B[D]:=\{$polynomials of $D$, with coefficients in $B$.$\}$ is a dense sub-algebra of $A$. The only question is I am not sure if it is proper. Is that a possible construction?

Answer 1

If $A/I$ is infinite dimensional, choose a proper dense sualgebra $D\subseteq A/I$ (can you prove that such a thing exists?) and let $E=q^{-1}(D)$. Then $E$ is a proper dense sualgebra containing $I$, and hence also $B$.