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A Constructive Proof of the Generalized Gelfand Isomorphism

Authors:	V M Buchstaber E G Rees

Institution:	(1) Department of Mathematics and Mechanics, Moscow State University, Russia;(2) Department of Mathematics and Statistics, Edinburgh University, UK

Abstract:	Using an analog of the classical Frobenius recursion, we define the notion of a Frobenius $\user1{n}$ -homomorphism. For $\user1{n} = 1$ , this is an ordinary ring homomorphism. We give a constructive proof of the following theorem. Let X be a compact Hausdorff space, $Sym^\user1{n} (X)$ the $\user1{n}$ th symmetric power of X, and $\mathbb{C}(M)$ the algebra of continuous complex-valued functions on X with the sup-norm; then the evaluation map ${\mathcal{E}}:Sym^\user1{n} (X) \to Hom(\mathbb{C}(X),\mathbb{C})$ defined by the formula $\user1{x}_1 , \ldots ,\user1{x}_\user1{n} ] \to (\user1{g} \to \sum {\user1{g}(} \user1{x}_\user1{k} ))$ identifies the space $Sym^\user1{n} (X)$ with the space of all Frobenius $\user1{n}$ -homomorphisms of the algebra $(\mathbb{C}(X)$ into $\mathbb{C}$ with the weak topology.

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