On a semigroup problem

Viorel Nitica,Andrei Török

doi:10.3934/dcdss.2019148

Abstract

If \begin{document}$ S $\end{document} is a semigroup in \begin{document}$ \mathbb{R}^n $\end{document} that is not separated by a linear functional, then it is known that the closure of \begin{document}$ S $\end{document} is a group. We investigate a similar statement in an infinite dimensional topological vector space \begin{document}$ X $\end{document} . We show that if \begin{document}$ X $\end{document} is an infinite dimensional Banach space, then there exists a semigroup \begin{document}$ S\subset X $\end{document} , not separated by the continuous functionals supported by the closed linear span of \begin{document}$ S $\end{document} , for which the closure of the semigroup is not a group. If \begin{document}$ X $\end{document} is an infinite dimensional Frechet space, then the closure of a semigroup that is not separated is always a group if and only if \begin{document}$ X $\end{document} is \begin{document}$ \mathbb{R}^{\omega} $\end{document} , the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as \begin{document}$ \mathbb{R}^{\infty} $\end{document} , the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.

Highlights

The topological vector spaces considered here are over the field R of real numbers
X contains a semigroup S that is not separated by any non-zero continuous linear functional and which does not contain 0 in its closure
A structure theorem of Bessaga, Peczynski and Rolewicz [4] shows that any Frechet space is either isomorphic to a product of a Banach space and Rω or contains a closed subspace which is topologically isomorphic to an infinite dimensional nuclear Frechet space with Schauder basis and a continuous norm

Summary

Introduction

The topological vector spaces considered here are over the field R of real numbers. The goal of this paper is to investigate the following basic problem: Semigroup Problem. Assume that S is not separated by any continuous linear functional in the dual of X0, that is, for any φ ∈ (X0)∗ \ {0} there exists x1, x2 ∈ S such that φ(x1) > 0 and φ(x2) < 0 Does it follow that the closure of S is a group? A continuous linear functional φ ∈ (Rω)∗ can be identified with a sequence of real numbers with finite support φ = (φn)∞ n=1, that is there exists an integer N (φ) such that φn = 0 if n > N (φ). Let S ⊂ Rω be a semigroup that is not separated by any non-zero continuous linear functional. Assume that there exists an increasing sequence (ni)∞ i=1 of positive integers such that Sni is not separated by any non-zero continuous linear functional for all i. We conclude that g−1 belongs to the closure of S

The case where X is a Banach space

The case where X is a Frechet space

Applications

Further results and additional directions of study