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 $f(p) \neq f(q)$.

For every $x\in X$ define $\varepsilon_x(f)=f(x),$ for $f\in A$. Then $\varepsilon_x$
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
* Graham Allan & H. Garth Dales (2011) ''Introduction to Banach Spaces and Algebras'', Oxford University Press

Banach algebras

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