= Boundary-analytic disk algebra with doubled character space
{title2=$\Phi_C\cong\overline{\mathbb D}\cup_{\partial\mathbb D}\overline{\mathbb D}\cong S^2$}
Let $C$ consist of continuous functions on the closed unit disk whose boundary values extend to a member of the <disk algebra>. The extension is unique by the <maximum modulus principle>; the extension map $T:C\to A(\overline{\mathbb D})$ is a surjective unital algebra homomorphism of norm at most one. Its kernel $I$ is the <ideal> of functions vanishing on the boundary. A character vanishing on $I$ factors through $T$, so it is $f\mapsto T(f)(\lambda)$ for some closed-disk point $\lambda$. For a character $\chi$ not vanishing on $I$, choose $h\in I$ with $\chi(h)\ne0$. The rule $F\mapsto\chi(hF)/\chi(h)$ extends $\chi$ to a character of all continuous functions on the disk: multiplicativity follows from $\chi(hF)\chi(hG)=\chi(h)\chi(hFG)$. It is therefore evaluation at an interior point, since its value at $h$ is nonzero. These two closed-disk families agree exactly on their boundaries. Their glued compact space maps continuously and bijectively to the Hausdorff <character space of an algebra>, giving the stated <homeomorphism>. The algebra is closed because uniform convergence of boundary data gives uniform convergence of their analytic extensions.
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