Disk algebra is a concept that arises in the area of functional analysis, particularly in the study of function spaces and operator theory. Specifically, it refers to the algebra of holomorphic (analytic) functions defined on the open unit disk in the complex plane. The disk algebra, often denoted as \( A(D) \), consists of all continuous functions on the closed unit disk that are holomorphic in the interior of the disk.
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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 converge uniformly to as , and each is uniformly approximated on the closed disk by its Taylor polynomials. A character of an algebra is therefore determined by its value at the coordinate function; boundedness gives , and polynomial approximation yields . Consequently the disk algebra is a natural uniform algebra on the closed disk.