Operators on superspaces and generalizations of the Gelfand–Kolmogorov theorem

Gelfand and Kolmogorov in 1939 proved that a compact Hausdorff topological space X can be canonically embedded into the infinite‐dimensional vector space ${\mathrm{C(X)}}^{*},$ the dual space of the algebra of continuous functions $\mathrm{C(X),}$ as an “algebraic variety”, specified by an infinite system of quadratic equations.

Buchstaber and Rees have recently extended this to all symmetric powers ${\mathrm{Sym}}^n\mathrm{(X)}$ 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 algebra.