A Constructive Proof of the Generalized Gelfand Isomorphism

Using an analog of the classical Frobenius recursion, we define the notion of a Frobenius $n$-homomorphism. For $n=1$, this is an ordinary ring homomorphism. We give a constructive proof of the following theorem. Let $X$ be a compact Hausdorff space, $\operatorname{Sym}^n(X)$ the $n$th symmetric power of $X$, and $\mathbb{C}(X)$ the algebra of continuous complex-valued functions on $X$ with the sup-norm; then the evaluation map $\mathcal{E}\colon\operatorname{Sym}^n(X)\to\operatorname{Hom}(\mathbb{C}(X),\mathbb{C})$ defined by the formula $[x_1,\dots,x_n]\to(g\to\sum g(x_k))$ identifies the space $\operatorname{Sym}^n(X)$ with the space of all Frobenius $n$-homomorphisms of the algebra $\mathbb{C}(X)$ into $\mathbb{C}$ with the weak topology.