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)$$
ef(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''.

Contents

Abstract characterization

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.