In functional analysis, 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 ∈ {\displaystyle \in } X there is f ∈ {\displaystyle \in } A with f(x) ≠ {\displaystyle \neq } 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 are precisely the ideals M x {\displaystyle M_{x}} of functions vanishing at a point x in X.
Abstract characterization
If A is a unital commutative Banach algebra such that | | a 2 | | = | | a | | 2 {\displaystyle ||a^{2}||=||a||^{2}} 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.
Notes
- Gamelin, Theodore W. (2005). Uniform Algebras. American Mathematical Soc. ISBN 978-0-8218-4049-8.
- Gorin, E.A. (2001) [1994], "Uniform algebra", Encyclopedia of Mathematics, EMS Press
References
(Gamelin 2005, p. 25) - Gamelin, Theodore W. (2005). Uniform Algebras. American Mathematical Soc. ISBN 978-0-8218-4049-8. ↩