Banach algebras | Functional analysis

Uniform algebra

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 Afor every x, y X there is fA 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 are precisely the ideals of functions vanishing at a point x in X. (Wikipedia).

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Related pages

Compact space | Topological space | Uniform norm | Hausdorff space | Functional analysis | Gelfand representation | Unital algebra | Banach function algebra | Algebra over a field | Maximal ideal | Banach algebra | Continuous function | C*-algebra | Closed set