= Disk algebra
{title2=$A(\overline{\mathbb D})$}
{wiki}
= Disc algebra
{synonym}
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 $r\uparrow1$, 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 $|\lambda|\leq1$, and polynomial approximation yields $\varphi(f)=f(\lambda)$. Consequently the <disk algebra> is a <natural uniform algebra> on the closed disk.
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