Dense subalgebra intersection with essential ideal is dense?

Question

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.

Answer 1

Certanily not. Consider $A=C([0, 1])$, $$B=\{f\in A: f(0)=0\},$$ and $D=\mathbb C\cdot 1$.

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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

1. $B\cap D$ 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.

2. $B$ 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].

3. $B$ 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.