Banach function algebra

In functional analysis a Banach function algebra on a compact Hausdorff space X is 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.

Theorem: Each commutative unital, semisimple Banach algebra (that is one with Jacobson radical equal to zero) 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).

An important special case of Banach function algebras are uniform algebras.