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In functional analysis, a Banach function algebra on a compact Hausdorff space X is unital subalgebra, A, of the commutative C*-algebra C 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 = 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 ≠ f {\displaystyle f\neq f}.
For every x ∈ X {\displaystyle x\in X} define ε x = f , {\displaystyle \varepsilon _{x}=f,} for f ∈ A {\displaystyle f\in A}. Then ε x {\displaystyle \varepsilon _{x}} is a homomorphism on A {\displaystyle A} , non-zero if A {\displaystyle A} does not vanish at x {\displaystyle x}.
Theorem: A Banach function algebra is semisimple and each commutative unital, semisimple Banach algebra is isomorphic to a Banach function algebra on its character space.