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)

\ne

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.

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