Uniform algebra

In functional analysis, a uniform algebra A on a compact Hausdorff topological space X is a closed subalgebra of the C*-algebra C with the following properties:
As a closed subalgebra of the commutative Banach algebra C a uniform algebra is itself a unital commutative Banach algebra. Hence, it is, a Banach function algebra.
A uniform algebra A on X is said to be natural if the maximal ideals of A are precisely the ideals of functions vanishing at a point x in X.

Abstract characterization

If A is a unital commutative Banach algebra such that for all a in A, then there is a compact Hausdorff X such that A is isomorphic as a Banach algebra to a uniform algebra on X. This result follows from the spectral radius formula and the Gelfand representation.