Uniform algebra

In functional analysis, a uniform algebra A on a compact Hausdorff 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 following properties:¹

the constant functions are contained in A

for every ${\displaystyle x,y\in X}$ there is ${\displaystyle f\in A}$ with ${\displaystyle f(x)\neq f(y)}$. This is called separating the points of X.

As a closed subalgebra of the commutative algebra (structure) Banach algebra C(X), a uniform algebra is itself a unital commutative Banach algebra (when equipped with the uniform norm). Hence, it is (by definition) 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 ${\displaystyle M_{x}}$ of functions vanishing at a point x in X.

If A is a unital commutative Banach algebra such that ${\displaystyle \|a^{2}\|=\|a\|^{2}}$ 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.