In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions
(where D is the open unit disk in the complex plane ) that extend to a continuous function on the closure of D. That is,
where denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space).
When endowed with the pointwise addition and pointwise multiplication this set becomes an algebra over C, since if f and g belong to the disk algebra, then so do f + g and fg.
Given the uniform norm
by construction, it becomes a uniform algebra and a commutative Banach algebra.
By construction, the disc algebra is a closed subalgebra of the Hardy space H<sup>∞</sup>. In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of Fatou that a general element of H<sup>∞</sup> can be radially extended to the circle almost everywhere.