Algebraic structure of function rings of some universal spaces

Using an algebraic characterisation of zero-dimensional mappings the author constructed universal compacts $Z(B,H)$ for the spaces possessing zero-dimensional mappings into the given compact $B$, where $H$ is a collection of functions on $B$ which separates points and closed subsets. By the characterisation theorem due to M. Bestvina for $B=S^n$ and an appropriate $H$ it is proved that the compact $Z(B,H)$ coincides with the Menger's universal compact $\mu^n$. As an application one gets a description of the ring $C_{\mathbb R}(\mu^n)$ as the closure of the polynomial ring $C_{\mathbb R}(S^n)[u_1,u_2,\dots,u_k,\dots]$ on elements $u_k$ such that $u_k^2=h_k^+$ for some $h_k^+\in C_{\mathbb R}(S^n)$. Another application is an representation of $\mu^n$ as the inverse limit of real algebraic manifolds. The complexification of this construction leads to some compact $E^{2n}$ which is the inverse limit of compactifications of complex algebraic manifolds without singularities and contains $\mu^n$ as the fixed set of the involution generated by the complex conjugation. On $E^{2n}$ an action of the countable product of order 2 cyclic groups is defined; the orbit-space of this action is a compactification of the tangent bundle $T(S^n)$.