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 → [math]\displaystyle{ \mathbb{C} }[/math],

(where D is the open unit disk in the complex plane [math]\displaystyle{ \mathbb{C} }[/math]) that extend to a continuous function on the closure of D. That is,

[math]\displaystyle{ A(\mathbf{D}) = H^\infty(\mathbf{D})\cap C(\overline{\mathbf{D}}), }[/math]

where H^∞(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,

[math]\displaystyle{ \|f\| = \sup\{|f(z)|\mid z\in \mathbf{D}\}=\max\{ |f(z)|\mid z\in \overline{\mathbf{D}}\}, }[/math]

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^∞. 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^∞ can be radially extended to the circle almost everywhere.