Dense subalgebra intersection with essential ideal is dense? Consider $A$ is an unital C* algebra, $D$ is a dense *-subalgebra, $B$ is a essential,maximal ideal of $A$. Is $D\cap B$ dense in $B$?
I have seen this question, and this question, but my assumption is little bit different, so I think I am asking a different question here.
 A: Certanily not.  Consider $A=C([0, 1])$,  $$B=\{f\in  A: f(0)=0\},$$ and $D=\mathbb C\cdot 1$.

EDIT.
OK, so here is a counter-example satisfying  the new requirement that $D$ be a dense subalgebra.
$\newcommand{\F}{\mathbb F_2}$
Let $\F$ be the free group on two generators, and let:

*

*$A$ be the full group C$^*$-algebra of $\F$, often also denoted by $C^*(\F)$,


*$B$ be the kernel of the left-regular representation $\lambda :C^*(\F)\to C_r^*(\F)$, and


*$D$ be the canonical copy of the complex group algebra $\mathbb C(G)$ within $C^*(\F)$.
Then

*

*$B\cap D$ is not dense in $B$,  and in fact  $B\cap D=\{0\}$.

Reason.  This is because the restriction of $\lambda $ to $\mathbb C(G)$ is injective.


*$B$ is a maximal ideal of $C^*(\F)$.

Reason.  This is because $C_r^*(\F)$ is simple,  thanks to a well known result by Powers [1].


*$B$ is an  essential ideal.

Reason.  This is the hardest point to justify:  arguing by contradiction,  suppose that $J$ is a nontrivial ideal in $C^*(\F)$,
trivially intersecting $B$.  Then $\lambda (J)$ is a nontrivial ideal in  $C_r^*(\F)$,  so necessarily $\lambda (J)=C_r^*(\F)$, by
simplicity.
It follows that $\lambda $ restricts to an isomprphism from $J$ to $C_r^*(\F)$, and hence  the inverse of that restriction is  an
injective *-homomorphism
$$
  \mu :C_r^*(\F)\to C^*(\F).
  $$
By Choi's Theorem [2], $C^*(\F)$ admits a separating family of finite dimensional representations (this is to say that  $C^*(\F)$ is
residually finite dimensional), so the same applies to $C_r^*(\F)$.  However, since the latter is simple and infinite
dimensional, we see that it has no finite dimensional representation whatsoever.  This is a contradiction.
[1] Powers, Robert T., Simplicity of the C$^*$-algebra associated with the free
group on two generators, Duke Math. J. 42, 151-156
(1975). ZBL0342.46046.
[2] Choi, Man-Duen, The full C$^*$-algebra of the free group on two
generators, Pac. J. Math. 87, 41-48
(1980). ZBL0463.46047.
