Disk algebra

Template:Short description 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,

$A(\mathbf{𝐃})=H^{\mathrm{\infty }}(\mathbf{𝐃})\cap C(\overline{\mathbf{𝐃}}),$

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

$\Vert f\Vert =\sup \{|f(z)|\mid z\in \mathbf{𝐃}\}=\max \{|f(z)|\mid z\in \overline{\mathbf{𝐃}}\},$

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^∞. 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.