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The Gelfand map and symmetric products

V. M. Buchstaber·E. G. Rees
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Abstract

If A is an algebra of functions on X, there are many cases when X can be regarded as included in Hom(A,C) as the set of ring homomorphisms. In this paper the corresponding results for the symmetric products of X are introduced. It is shown that the symmetric product Sym^n(X) is included in Hom(A,C) as the set of those functions that satisfy equations generalising f(xy)=f(x)f(y). These equations are related to formulae introduced by Frobenius and, for the relevant A, they characterise linear maps on A that are the sum of ring homomorphisms. The main theorem is proved using an identity satisfied by partitions of finite sets.

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V. M. Buchstaber, E. G. Rees. 2001-09-18. The Gelfand map and symmetric products. https://arxiv.org/abs/math/0109122

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