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 $f\in A$ {\displaystyle f\in A}. A function algebra separates points if for each distinct pair of points $p,q\in X$ {\displaystyle p,q\in X}, there is a function $f\in A$ {\displaystyle f\in A} such that $f(p)\neq f(q)$ {\displaystyle f(p)\neq f(q)}.

For every $x\in X$ {\displaystyle x\in X} define $\varepsilon _{x}(f)=f(x),$ {\displaystyle \varepsilon _{x}(f)=f(x),} for $f\in A$ {\displaystyle f\in A}. Then $\varepsilon _{x}$ {\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.

References

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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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v t e Spectral theory and ^*-algebras
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