# 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* which 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 . Then is a non-zero homomorphism (character) on .

**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.

## References

- H.G. Dales
*Banach algebras and automatic continuity*