Ob obobshchennykh simmetricheskikh stepeniakh i obobshchenii teori Kolmogorova-Gel'fanda-Bukhshtabera-Risa (On generalized symmetric powers and a generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory)

The classical Kolmogorov-Gelfand theorem gives an embedding of a (compact Hausdorff) topological space X into the linear space of all linear functionals C(X)^* on the algebra of continuous functions C(X). The image is specified by the algebraic equations f(ab)=f(a)f(b) for all functions a, b on X; that is, the image consists of all algebra homomorphisms of C(X) to numbers. Buchstaber and Rees have found that not only X, but all symmetric powers of X can be embedded into the space C(X)^*. The embedding is again given by algebraic equations, but more complicated. The algebra homomorphisms are replaced by the so-called n-homomorphisms, the notion that can be traced back to Frobenius, but which explicitly appeared in Buchstaber and Rees's works on multivalued groups. We give a further generalization of the Kolmogorov-Gelfand-Buchstaber-Rees theory. The symmetric powers of a space X or of an algebra A are replaced by certain "generalized symmetric powers" Sym^{p|q}(X) and S^{p|q}A, respectively, and the n-homomorphisms, by the new notion of "p|q-homomorphisms". An important tool of our study is a certain "characteristic function" R(f,a,z), for an arbitrary linear map of algebras f: A\to B. The functional properties with respect to the variable z reflect algebraic properties of the map f.