A Constructive Proof of the Generalized Gelfand Isomorphism

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.