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 that makes it a Banach algebra.
A function algebra is said to vanish at a point p if f(p) = 0 for all . A function algebra separates points if for each distinct pair of points , there is a function such that .
For every define for . Then is a homomorphism (character) on , non-zero if does not vanish at .
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 is the uniform norm (or sup-norm) on , then is called a uniform algebra. Uniform algebras are an important special case of Banach function algebras.