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 that makes it a Banach algebra.

A function algebra is said to vanish at a point p if f(p) = 0 for all ${\displaystyle f\in A}$. A function algebra separates points if for each distinct pair of points ${\displaystyle p,q\in X}$, there is a function ${\displaystyle f\in A}$ such that ${\displaystyle f(p)\neq f(q)}$.

For every ${\displaystyle x\in X}$ define ${\displaystyle \varepsilon _{x}(f)=f(x),}$ for ${\displaystyle f\in A}$. Then ${\displaystyle \varepsilon _{x}}$is a homomorphism (character) on ${\displaystyle A}$, non-zero if ${\displaystyle A}$ does not vanish at ${\displaystyle x}$.

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 ${\displaystyle A}$ is the uniform norm (or sup-norm) on ${\displaystyle X}$, then ${\displaystyle A}$ is calleda uniform algebra. Uniform algebras are an important special case of Banach function algebras.