Operators on superspaces and generalizations of the Gelfand-Kolmogorov theorem

(Write-up of a talk at the Bialowieza meeting, July 2007.) Gelfand and Kolmogorov in 1939 proved that a compact Hausdorff topological space $X$ can be canonically embedded into the infinite-dimensional vector space $C(X{)}^{\ast }$, the dual space of the algebra of continuous functions $C(X)$ as an “algebraic variety” specified by an infinite system of quadratic equations. Buchstaber and Rees have recently extended this to all symmetric powers ${\text{Sym}}^n(X)$ using their notion of the Frobenius $n$-homomorphisms. We give a simplification and a further extension of this theory, which is based, rather unexpectedly, on results from super linear lgebra.