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 algebrai ...
, 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 deriv ...
s
:''ƒ'' : D →
,
(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 di ...
in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
) that extend to a continuous function on the
closure of D. That is,
:
where denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain 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 ...
).
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
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
over C, since if ''ƒ'' and ''g'' belong to the disk algebra then so do ''ƒ'' + ''g'' and ''ƒg''.
Given the
uniform norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when t ...
,
:
by construction it becomes a
uniform algebra
In functional analysis, a uniform algebra ''A'' on a compact Hausdorff topological space ''X'' is a closed (with respect to the uniform norm) subalgebra of the C*-algebra ''C(X)'' (the continuous complex-valued functions on ''X'') with the fol ...
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
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain 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 ...
. 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
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