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 ∈ A {\displaystyle f\in A} . A function algebra separates points if for each distinct pair of points p , q ∈ X {\displaystyle p,q\in X} , there is a function f ∈ A {\displaystyle f\in A} such that f ( p ) ≠ f ( q ) {\displaystyle f(p)\neq f(q)} .
For every x ∈ X {\displaystyle x\in X} define ε x ( f ) = f ( x ) , {\displaystyle \varepsilon _{x}(f)=f(x),} for f ∈ A {\displaystyle f\in A} . Then ε x {\displaystyle \varepsilon _{x}} is a homomorphism (character) on A {\displaystyle A} , non-zero if A {\displaystyle A} does not vanish at x {\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 A {\displaystyle A} is the uniform norm (or sup-norm) on X {\displaystyle X} , then A {\displaystyle A} is called a uniform algebra. Uniform algebras are an important special case of Banach function algebras.
- Andrew Browder (1969) Introduction to Function Algebras, W. A. Benjamin
- H.G. Dales (2000) Banach Algebras and Automatic Continuity, London Mathematical Society Monographs 24, Clarendon Press ISBN 0-19-850013-0
- Graham Allan & H. Garth Dales (2011) Introduction to Banach Spaces and Algebras, Oxford University Press ISBN 978-0-19-920654-4