Banach function algebra explained

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) ≠ 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 calleda uniform algebra. Uniform algebras are an important special case of Banach function algebras.

References

• Andrew Browder (1969) Introduction to Function Algebras, W. A. Benjamin