In mathematics, specifically in functional and

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

, the disk algebra ''A''(D) (also spelled disc algebra) is the set of

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

s : ''ƒ'' : D →

\mathbb

(where D is the

open unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...

in the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

\mathbb

) that extend to a continuous function on the closure of D. That is, :

A(\mathbf) = H^\infty(\mathbf) \cap C(\overline),

where denotes the

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

of bounded analytic functions on the unit disc D (i.e. a

Hardy space In complex analysis, the Hardy spaces (or Hardy classes) H^p are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real anal ...

). When endowed with the pointwise addition and pointwise multiplication this set becomes an

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

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\big\ = \max\big\,

by construction, it becomes a uniform algebra and a commutative

Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...

. By construction, the disc algebra is a closed subalgebra of the

''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 In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

References

Functional analysis Complex analysis Banach algebras {{mathanalysis-stub