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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 <math>\in</math> X there is f<math>\in</math>A with f(x)<math>\ne</math>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 a itself a commutative Banach algebra (when equipt with the uniform norm).

A uniform algebra A on X is said to be natural if the maximal ideals of A precisely are the ideals <math>M_x</math> of functions vanishing at a point x in X