A constructive proof of the generalized Gelfand isomorphism (Q5959521)

Let \(X\) be a compact Hausdorff space and \(C(X)\) denote the algebra of all continuous mappings from \(X\) into the complex numbers \(C\) with the sup norm. Let \(\text{Hom}(C(X),C)\) be the space of all continuous linear functionals from \(C(X)\) into \(C\) with the weak topology. The evaluation map \(e\) sends each \(x_1\in X\) to the element \(f\in \text{Hom}(C(X),C)\) defined by \(f(g)= g(x_1)\) for all \(g\in C(X)\). A result of Gelfand says that \(e\) maps \(X\) homeomorphically onto \(\text{Hom}(C(X),C)\). In an earlier \(e\)-print (uk.arXiv.org:mathC0/0109122) the present authors extended this result by defining inductively a Frobenius \(n\)-homomorphism \(f\) from \(C(X)\) into \(C\), replacing \(X\) with its \(n\)th symmetric product \(\text{Sym}^n(X)\) and defining \(e(x_1,\dots, x_n)= f\) where \(f(g)= g(x_1)+\cdots+ g(x_n)\). In the present paper, they show how to construct the set \((x_1,\dots, x_n)\) for any given \(f\).