In functional analysis, 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:[1] 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.
of functions vanishing at a point x in X.
If A is a unital commutative Banach algebra such that
This result follows from the spectral radius formula and the Gelfand representation.