On Valdivia-type universal topological algebras
by
Leonid Shapiro
Academy of Labor and Social Relations, ul. Lobachevskogo 90, 119454 Moscow, Russia

We start with the following two classical results of topological algebra.

Theorem   (R.J. Nunke, 1962). If G is a closed subgroup of Z^w, then G is isometric ( ≡ isomorphic and homeomorphic) to a power of Z.

Theorem   (S.U. Chase, 1963). Let R be a principal ideal domain equipped with the discrete topology. If M is a closed submodule (over R) of R^w, then M is isometric to a power of R.

For the generalization of these results we need the following

Definition.   Let X=∏{ (X_s, x_s^*): s ∈ S} be the product of a family of pointed topological spaces and S ⊂ X is the usual S-product. A closed subspace Y ⊂ X is called a Valdivia-type subspace if Y∩S is everywhere dense in Y.

The main results of the talk are the following

Theorem   (Discrete case). Let R be a principal ideal domain equipped with the discrete topology. If M is a Valdivia-type submodule (over R) of R^t, then M is isometric to a power of R.

Theorem   (Nondiscrete case). Let K be a complete nondiscrete normed division ring. If E is a Valdivia-type subspace (over K) of K^t, then E is isometric to a power of K.

In the talk we generalize these results to the case of universal topological algebras.

Date received: June 25, 2008