# 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.

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

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.

1. $$B$$ 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].

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.

• May you elaborate how is $B$ essential? Feb 24, 2021 at 18:32
• Sorry but I have edited the $D$ must be dense right now. Feb 24, 2021 at 18:39