Banach function algebra

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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(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  (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) \neq f(q) .

For every x\in X define \varepsilon_x(f)=f(x)\ (f\in 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 called a uniform algebra. Uniform algebras are an important special case of Banach function algebras.

References[edit]

  • H.G. Dales Banach algebras and automatic continuity