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Uniform Algebra
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(Definition)
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In what follows we will show that the Gelfand transform $\Gamma_{\mathcal{A}} \colon \mathcal{A} \mapsto \hat{\mathcal{A}}$ of a commutative, unital Banach algebra $\mathcal{A}$ is an isometry if and only if $\mathcal{A}$ is a uniform Banach algebra.
Denote by $M(\mathcal{A})$ the space of (continuous) characters on $\mathcal{A}$ . Recall that for all $f \in \mathcal{A}$ the spectrum $\sigma(f)$ of $f$ is identical with the range $\hat{f}(M(\mathcal{A}))$ and the spectral radius $r(f) = \|\hat{f}\|_{M(\mathcal{A})} = \lim_{n \to \infty}
\|f^n\|^{\frac{1}{n}}$
Proposition 1. A Banach algebra $\mathcal{A}$ is uniform if and only if it's Gelfand transform $\Gamma_{\mathcal{A}} \colon \mathcal{A} \to \hat{\mathcal{A}}, f \mapsto \hat{f}$ is isometric.
Proof. If $f \mapsto \hat{f}$ is an isometry we have $\|f^2\| = \|\hat{f}^2\|_{M(\mathcal{A})} = \|\hat{f}\|_{M(\mathcal{A})}^2 = \|f\|^2$ .
Conversely assume $\|f^2\| = \|f\|^2$ for all $f \in \mathcal{A}$ . Then by induction we have $\|f^{2^n}\| = \|f\|^{2^n}$ for all $n \in \mathbb{N}$ . Hence $\|f\| = \|f^{2^n}\|^{\frac{1}{2^n}} \to \|\hat{f}\|_{M(\mathcal{A})}$ . 
The following characterization is also often given as the definition of a uB algebra.
Proposition 2. A Banach algebra $\mathcal{A}$ is uniform iff it is topologically and algebraically isomorphic to a closed, pointseparating subalgebra of $C(X)$ for $X$ a compact Hausdorff space.
Proof. Since $\hat{\mathcal{A}}$ separates the points of the compact, nonempty space $M(\mathcal{A})$ we see that a uB algebra $\mathcal{A}$ must have this property.
Conversely, let $\mathcal{A}$ be a closed pointseparating subalgebra of $C(X)$ . Then clearly $\|f\|_X^2 = \|f^2\|_X$ for all $f \in \mathcal{A}$ . 
- Theodore W. Gamelin Uniform Algebras, Oxford University Press, New Edition, 2005
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"Uniform Algebra" is owned by karstenb.
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| Keywords: |
uB algebra, uniform algebra |
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Cross-references: property, points, Hausdorff space, compact, subalgebra, closed, isomorphic, iff, characterization, induction, conversely, isometric, proposition, spectral radius, range, spectrum, characters, continuous, isometry, Gelfand transform, algebra, Banach algebra, unital, commutative
This is version 2 of Uniform Algebra, born on 2009-10-18, modified 2009-12-28.
Object id is 11957, canonical name is UniformAlgebra.
Accessed 286 times total.
Classification:
| AMS MSC: | 46J10 (Functional analysis :: Commutative Banach algebras and commutative topological algebras :: Banach algebras of continuous functions, function algebras) | | | 46J40 (Functional analysis :: Commutative Banach algebras and commutative topological algebras :: Structure, classification of commutative topological algebras) |
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Pending Errata and Addenda
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