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| In [[functional analysis]] a '''Banach function algebra''' on a [[compact space|compact]] [[Hausdorff space]] ''X'' is [[unital algebra|unital]] [[subalgebra]], ''A'' of the [[commutative]] [[C*-algebra]] ''C(X)'' of all [[continuous function|continuous]], [[complex number|complex]] valued functions from ''X'', together with a norm on ''A'' which makes it a [[Banach algebra]].
| | Greetings. The writer's title is Phebe and she feels comfortable when people use the full name. California is our birth location. He is really fond of performing ceramics but he is struggling to discover time for it. In her professional lifestyle she is a payroll clerk but she's usually needed her personal business.<br><br>My website - [http://vine.ac/xe/?document_srl=482964 vine.ac] |
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| A function algebra is said to vanish at a point p if f(p) = 0 for all <math> (f\in A) </math>. A function algebra separates points if for each distinct pair of points <math> (p,q \in X) </math>, there is a function <math> (f\in A) </math> such that <math> f(p) \neq f(q) </math>.
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| For every <math>x\in X</math> define <math>\varepsilon_x(f)=f(x)\ (f\in A)</math>. Then <math>\varepsilon_x</math>
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| is a non-zero homomorphism (character) on <math>A</math>. | |
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| '''Theorem:''' A Banach function algebra is [[semisimple algebra|semisimple]] (that is its [[Jacobson radical]] is equal to zero) and each commutative [[unital ring|unital]], semisimple Banach algebra is [[isomorphic]] (via the [[Gelfand transform]]) to a Banach function algebra on its [[character space]] (the space of algebra homomorphisms from ''A'' into the complex numbers given the [[relative topology|relative]] [[weak* topology]]).
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| If the norm on <math>A</math> is the uniform norm (or sup-norm) on <math>X</math>, then <math>A</math> is called
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| a '''uniform algebra'''. Uniform algebras are an important special case of Banach function algebras.
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| ==References== | |
| * H.G. Dales ''Banach algebras and automatic continuity''
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| {{Mathanalysis-stub}}
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| [[Category:Banach algebras]]
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Latest revision as of 05:23, 10 October 2014
Greetings. The writer's title is Phebe and she feels comfortable when people use the full name. California is our birth location. He is really fond of performing ceramics but he is struggling to discover time for it. In her professional lifestyle she is a payroll clerk but she's usually needed her personal business.
My website - vine.ac