Banach function algebra

In functional analysis a Banach function algebra on a compact Hausdorff space X is unital subalgebra, A of the commutative C*-algebra C(X) of all continuous, complex valued functions from X, together with a norm on A which makes it a Banach algebra.

A function algebra is said to vanish at a point p if f(p) = 0 for all ( f ∈ A ) . A function algebra separates points if for each distinct pair of points ( p , q ∈ X ) , there is a function ( f ∈ A ) such that f ( p ) ≠ f ( q ) .

For every x ∈ X define ε x ( f ) = f ( x )   ( f ∈ A ) . Then ε x is a non-zero homomorphism (character) on A .

Theorem: A Banach function algebra is semisimple (that is its Jacobson radical is equal to zero) and each commutative unital, semisimple Banach algebra is isomorphic (via the Gelfand transform) to a Banach function algebra on its character space (the space of algebra homomorphisms from A into the complex numbers given the relative weak* topology).

If the norm on A is the uniform norm (or sup-norm) on X , then A is called a uniform algebra. Uniform algebras are an important special case of Banach function algebras.