Disk algebra
In functional and complex analysis, the disk algebra A is the set of holomorphic functions
where D is the open unit disk in the complex plane C, f extends 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.
When endowed with the pointwise addition f+g, and pointwise multiplication f'g, this set becomes an algebra over C', since if ƒ and g belong to the disk algebra then so do ƒ + g and ƒg.
Given the uniform norm,
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 infinity|. 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.
