Uniform algebra
{{Short description|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:{{harv|Gamelin|2005|p=25}}
: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 are precisely 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.
Notes
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References
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- {{cite book |isbn=978-0-8218-4049-8|title=Uniform Algebras |last1=Gamelin |first1=Theodore W. |year=2005 |publisher=American Mathematical Soc. }}
- {{Eom| title = Uniform algebra | author-last1 = Gorin| author-first1 = E.A.|
| oldid = 49068}}
{{ref end}}
{{Functional analysis}}
{{SpectralTheory}}
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