Английская Википедия: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 that makes it a Banach algebra.

A function algebra is said to vanish at a point p if f(p) = 0 for all <math> f\in A </math>. A function algebra separates points if for each distinct pair of points <math> p,q \in X </math>, there is a function <math> f\in A </math> such that <math> f(p) \neq f(q) </math>.

For every <math>x\in X</math> define <math>\varepsilon_x(f)=f(x),</math> for <math>f\in A</math>. Then <math>\varepsilon_x</math> is a homomorphism (character) on <math>A</math>, non-zero if <math>A</math> does not vanish at <math>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>A</math> is the uniform norm (or sup-norm) on <math>X</math>, then <math>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 Шаблон:ISBN
• Graham Allan & H. Garth Dales (2011) Introduction to Banach Spaces and Algebras, Oxford University Press Шаблон:ISBN

Шаблон:Mathanalysis-stub