Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense

If $X=\mathrm{\varnothing }$, then $C(X,\mathbb{C})$ has only the empty function, which is the zero element of $A$, so the full-closure conclusion holds.

Assume $X\mathrm{\ne }\mathrm{\varnothing }$. By [L1] and [L2], the real-valued part $A_{\mathbb{R}}$ either is dense in $C(X,\mathbb{R})$ or has a unique common zero $x_0$ and closure equal to the real functions vanishing there.

In the dense alternative, let $F\in C(X,\mathbb{C})$ and $\epsilon >0$; the coordinate functions $\mathrm{Re}F$ and $\mathrm{Im}F$ are continuous by [L5] and [L6], so choose $u,v\in A_{\mathbb{R}}$ within $\epsilon \mathrm{/}2$ of them and put $h:=u+iv\in A$.

In the common-zero alternative, [L1] says that the same unique $x_0$ is the common zero of $A$. Every uniform limit of members of $A$ vanishes at $x_0$, so $\bar{A}\subseteq I_{x_0}^{\mathbb{C}}$.

For every $x\in X$, [L4] gives $\mathrm{\mid }F(x)-h(x)\mathrm{\mid }\le \mathrm{\mid }\mathrm{Re}F(x)-u(x)\mathrm{\mid }+\mathrm{\mid }\mathrm{Im}F(x)-v(x)\mathrm{\mid }<\epsilon$, so $A$ is dense in $C(X,\mathbb{C})$.

Conversely, let $F\in I_{x_0}^{\mathbb{C}}$ and let $\epsilon >0$. Both $\mathrm{Re}F$ and $\mathrm{Im}F$ vanish at $x_0$; by the real alternative in step 1.2 they can be approximated within $\epsilon \mathrm{/}2$ by $u,v\in A_{\mathbb{R}}$, and the argument of step 3.1 puts $u+iv\in A$ within $\epsilon$ of $F$. As $\epsilon$ was arbitrary, $I_{x_0}^{\mathbb{C}}\subseteq \bar{A}$.

Steps 2.2, 3.1, and 4.1 transfer both real alternatives to $A$. If $A$ is unital, it contains the constant-one function and therefore has no common zero, so only the dense alternative is possible.