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.