Uniform algebra ($A\subseteq C(K)$)

A uniform algebra on a compact Hausdorff space $K$ is a closed complex subalgebra of $C(K)$, with the supremum norm, containing the constants and separating points. Evaluation embeds $K$ continuously and injectively in the character space of an algebra. The embedding is a homeomorphism onto a closed subset because its domain is compact and its target is Hausdorff. Extra characters need not be point evaluations on the specified $K$.

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 $\chi (h)\ne 0$. 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.

The disk algebra consists of functions continuous on the closed unit disk and holomorphic in its interior, with the supremum norm. Polynomials are dense: dilations $f_r(z)=f(rz)$ converge uniformly to $f$ as $r\uparrow 1$, and each $f_r$ is uniformly approximated on the closed disk by its Taylor polynomials. A character of an algebra is therefore determined by its value $\lambda$ at the coordinate function; boundedness gives $|\lambda |\le 1$, and polynomial approximation yields $\phi (f)=f(\lambda )$. Consequently the disk algebra is a natural uniform algebra on the closed disk.

A uniform algebra on $K$ is natural if every character of an algebra is evaluation at a point of $K$. The evaluation embedding then identifies its character space homeomorphically with $K$. A natural uniform algebra is characterized by the finite-generator criterion for a natural uniform algebra. In particular, $C(K)$ is natural; the disk algebra is natural when its underlying space is the closed disk.

A uniform algebra is natural exactly when functions with no common zero generate the unit ideal. If a generated ideal is proper, it is contained in a maximal ideal. That maximal ideal is closed: the closure of a proper ideal stays proper, since an element within distance less than one of the identity is invertible by the Neumann series. The Gelfand-Mazur theorem identifies the maximal-ideal quotient with the complex numbers, yielding a character of an algebra. Naturality would make that character a common-zero evaluation, a contradiction. Conversely, the asserted unit-ideal property makes the zero sets of every finite collection in a character's kernel intersect. Compactness gives a point common to all these zero sets; applying this to $f-\phi (f)1$ proves $\phi (f)=f(x)$. For $C(K)$, one can directly take $g_j=\overline{f_j}/{\sum }_k|f_k{|}^2$.