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 that makes it a Banach algebra. A function algebra is said to vanish at a point p if f(p) = 0 for all [math]\displaystyle{ f\in A }[/math]. A function algebra separates points if for each distinct pair of points [math]\displaystyle{ p,q \in X }[/math], there is a function [math]\displaystyle{ f\in A }[/math] such that [math]\displaystyle{ f(p) \neq f(q) }[/math].
For every [math]\displaystyle{ x\in X }[/math] define [math]\displaystyle{ \varepsilon_x(f)=f(x), }[/math] for [math]\displaystyle{ f\in A }[/math]. Then [math]\displaystyle{ \varepsilon_x }[/math] is a homomorphism (character) on [math]\displaystyle{ A }[/math], non-zero if [math]\displaystyle{ A }[/math] does not vanish at [math]\displaystyle{ x }[/math].
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 [math]\displaystyle{ A }[/math] is the uniform norm (or sup-norm) on [math]\displaystyle{ X }[/math], then [math]\displaystyle{ A }[/math] is called a uniform algebra. Uniform algebras are an important special case of Banach function algebras.
References
- 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
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