Operators on superspaces and generalizations of the Gelfand-Kolmogorov theorem

TL;DR

This paper explores the extension of the Gelfand-Kolmogorov theorem to symmetric powers of compact spaces, utilizing super linear algebra to simplify and generalize the existing framework.

Contribution

It introduces a simplified and extended approach to embedding symmetric powers of spaces into duals of function algebras, based on super linear algebra techniques.

Findings

Extended Gelfand-Kolmogorov theorem to symmetric powers

Utilized super linear algebra for simplification

Provided a new framework for algebraic embeddings

Abstract

(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)^* $, 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 $\Sym^n(X)$ 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.