Uniform algebra

A uniform algebra A on a compact Hausdorff topological space X is a closed subalgebra of the C*-algebra C(X) (the continuous complex valued functions on X) such that:

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

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