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 $$ f\in A $$. A function algebra separates points if for each distinct pair of points $$ p,q \in X $$, there is a function $$ f\in A $$ such that $$ f(p) \neq f(q) $$.

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

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 $$A$$ 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.