Uniform algebra

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 following properties:

the constant functions are contained in A

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

As a closed subalgebra of the commutative 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 precisely are the ideals $M_{x}$ of functions vanishing at a point x in X.

Abstract characterization

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