The Gelfand map and symmetric products

There are many instances of the principle that if A is an algebra of functions on X, then every ring homomorphism A → C is given by evaluation at a particular x ∈ X. Examples are the nullstellensatz in algebraic geometry and the result of Gelfand when X is a compact Hausdorff space. In these cases one can regard X as being included in Hom(A, C) as the set of those f : A → C which satisfy the set of equations f (xy) = f (x)f (y) indexed by (x, y) ∈ X × X. In this paper we introduce the corresponding equations for the symmetric products of X. We show that, in these examples, Sym n (X) is included in Hom(A, C) as the set of those f that satisfy these more complicated equations. Given a linear map f : A → C we consider certain maps which can be regarded as " higher " versions of f and are denoted Φ n (f) : A ⊗n → C, their definition is based on formulae used by G. Frobenius [Fro1]. The subset Φ n (A) ⊂ Hom(A, C) of the space of all linear maps Hom(A, C) consisting of those f for which Φ n+1 (f) = 0 and f (1) = n is particularly interesting and we will develop its properties; Φ 1 (A) is the set of ring homomorphisms. When A is an algebra of functions on a space X the sets Φ n (A) are closely related to the symmetric product Sym n (X) := X n /Σ n. The case n = 1 is classical, if A is a separating algebra of functions on a compact space X then Φ 1 (A) is the set of algebra homomorphisms and, by the Gelfand transform, this is homeomorphic to X. In many cases, we can identify Φ n (A) with the set of maps that can be written as the sum of n ring homomorphisms. In particular, when X a compact Hausdorff space and A = C(X) is the ring of complex valued continuous functions on X, Φ n (A) is precisely the set of linear maps that can be written as the sum of n ring homomorphisms and so can be identified with Sym n (X). The analogous result holds when A is a finitely generated commutative algebra. In particular, when X = C m …