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 $f\in A$. 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 $f(p)\ne f(q)$.

For every $x\in X$ define ${\epsilon }_x(f)=f(x),$ for $f\in A$. Then ${\epsilon }_x$ is a homomorphism (character) on $A$, non-zero if $A$ does not vanish at $x$.

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 $A$ 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

• 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 Template:ISBN
• Graham Allan & H. Garth Dales (2011) Introduction to Banach Spaces and Algebras, Oxford University Press Template:ISBN

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