# Uniform algebra

A **uniform algebra** *A* on a compact Hausdorff topological space *X* is a closed (with respect to the uniform norm) subalgebra of the C*-algebra *C(X)* (the continuous complex valued functions on *X*) with the following properties:

- the constant functions are contained in
*A* - for every
*x*,*y**X*there is f*A*with f(x)f(y). This is called separating the points of*X*.

As a closed subalgebra of the commutative Banach algebra *C(X)* a uniform algebra is itself a unital commutative Banach algebra (when equipped with the uniform norm). Hence, it is, (by definition) a Banach function algebra.

A uniform algebra *A* on *X* is said to be **natural** if the maximal ideals of *A* precisely are the ideals of functions vanishing at a point *x* in *X*.

## Abstract characterization

If *A* is a unital commutative Banach algebra such that for all *a* in *A*, then there is a compact Hausdorff *X* such that *A* is isomorphic as a Banach algebra to a uniform algebra on *X*. This result follows from the spectral radius formula and the Gelfand representation.