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:[1]
- the constant functions are contained in A
- for every x, y [math]\displaystyle{ \in }[/math] X there is f[math]\displaystyle{ \in }[/math]A with f(x)[math]\displaystyle{ \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 ideals of A are precisely the ideals [math]\displaystyle{ M_x }[/math] of functions vanishing at a point x in X.
Abstract characterization
If A is a unital commutative Banach algebra such that [math]\displaystyle{ ||a^2|| = ||a||^2 }[/math] 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
- ↑ (Gamelin 2005)
References
- Gamelin, Theodore W. (2005). Uniform Algebras. American Mathematical Soc.. ISBN 978-0-8218-4049-8.
- Hazewinkel, Michiel, ed. (2001), "Uniform algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
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