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 → C {\displaystyle \mathbb {C} }(where D is the open unit disk in the complex plane C {\displaystyle \mathbb {C} } ) that extend to a continuous function on the closure of D. That is,
A ( D ) = H ∞ ( D ) ∩ C ( D ¯ ) , {\displaystyle A(\mathbf {D} )=H^{\infty }(\mathbf {D} )\cap C({\overline {\mathbf {D} }}),}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 (f + g)(z) = f(z) + g(z) and pointwise multiplication (fg)(z) = f(z)g(z), 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
‖ f ‖ = sup { | f ( z ) | ∣ z ∈ D } = max { | f ( z ) | ∣ z ∈ D ¯ } , {\displaystyle \|f\|=\sup {\big \{}|f(z)|\mid z\in \mathbf {D} {\big \}}=\max {\big \{}|f(z)|\mid z\in {\overline {\mathbf {D} }}{\big \}},}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.