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)$e$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.