= Finite-generator criterion for a natural uniform algebra
{title2=$\bigcap_j f_j^{-1}(0)=\varnothing\Longrightarrow\sum_j f_jg_j=1$}
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 $f-\varphi(f)1$ proves $\varphi(f)=f(x)$. For $C(K)$, one can directly take $g_j=\overline{f_j}/\sum_k|f_k|^2$.
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