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.

For every $xin\; X$ define $varepsilon\_x(f)=f(x)\; (fin\; A)$. Then $varepsilon\_x$is a non-zero homomorphism (character) on $A$.

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

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