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 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 = 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 on, non-zero if does not vanish at.
Theorem: A Banach function algebra is semisimple and each commutative unital, semisimple Banach algebra is isomorphic to a Banach function algebra on its character space.
If the norm on is the uniform norm on, then is called
a uniform algebra. Uniform algebras are an important special case of Banach function algebras.