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A '''uniform algebra''' ''A'' on a [[compact space|compact]] [[Hausdorff space |Hausdorff]] [[topological space]] ''X'' is a closed (with respect to the [[uniform norm]]) [[algebra over a field|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'' <math>\in</math> ''X'' there is f<math>\in</math>''A'' with f(x)<math>\ne</math>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 ideal]]s of ''A'' precisely are the ideals <math>M_x</math> of functions vanishing at a point ''x'' in ''X''. | |||
==Abstract characterization== | |||
If ''A'' is a [[unital algebra|unital]] [[commutative]] [[Banach algebra]] such that <math>||a^2|| = ||a||^2</math> for all ''a'' in ''A'', then there is a [[compact space|compact]] [[Hausdorff space |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. | |||
{{mathanalysis-stub}} | |||
[[Category:Functional analysis]] | |||
[[Category:Banach algebras]] | |||
Revision as of 18:56, 13 December 2013
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 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 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.
