Disk algebra

In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions


 * ƒ : D → $$\mathbb{C}$$,

(where D is the open unit disk in the complex plane $$\mathbb{C}$$) that extend to a continuous function on the closure of D. That is,
 * $$A(\mathbf{D}) = H^\infty(\mathbf{D})\cap C(\overline{\mathbf{D}}),$$

where $H^{&infin;}(D)$ denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space). When endowed with the pointwise addition (ƒ + g)(z) = ƒ(z) + g(z), and pointwise multiplication (ƒg)(z) = ƒ(z)g(z), this set becomes an algebra over C, since if ƒ and g belong to the disk algebra then so do ƒ + g and ƒg.

Given the uniform norm,
 * $$\|f\| = \sup\{|f(z)|\mid z\in \mathbf{D}\}=\max\{ |f(z)|\mid z\in \overline{\mathbf{D}}\},$$

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 $H^{&infin;}$. 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&infin; can be radially extended to the circle almost everywhere.