Uniform algebra

Mathematical concept

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:^[1]

the constant functions are contained in A

for every x, y

\in

{\displaystyle \in } X there is f

\in

{\displaystyle \in }A with f(x)

\neq

{\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

[edit ]

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

[edit ]

^ (Gamelin 2005, p. 25)

References

[edit ]

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

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