# Uniform algebra

for every x, y ${\displaystyle \in }$ X there is f${\displaystyle \in }$A with f(x)${\displaystyle \neq }$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 ${\displaystyle M_{x}}$ 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.