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>\mathbb{C}</math>
(where D is the open unit disk in the complex plane <math>\mathbb{C}</math>) that extend to a continuous function on the closure of D. That is,
- <math>A(\mathbf{D}) = H^\infty(\mathbf{D}) \cap C(\overline{\mathbf{D}}),</math>
where Template:Math denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space).
When endowed with the pointwise addition Template:Nobr and pointwise multiplication Template:Nobr 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
- <math>\|f\| = \sup\big\{|f(z)| \mid z \in \mathbf{D}\big\} = \max\big\{|f(z)| \mid z \in \overline{\mathbf{D}}\big\},</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.