Digital comparator: Difference between revisions

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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:
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: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]]

Latest revision as of 17:31, 15 December 2014

Hi there, I am Alyson Pomerleau and I think it sounds quite good when you say it. For a while I've been in Mississippi but now I'm considering other options. The preferred hobby for him and his kids is to perform lacross and he would never give it up. He is an information officer.

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