Reflexivity in precompact groups and extensions

TL;DR

This paper investigates the conditions under which precompact and P-groups are reflexive, providing new criteria, counterexamples, and exploring how extensions affect reflexivity in these groups.

Contribution

It establishes new necessary and sufficient conditions for reflexivity in precompact Abelian groups and analyzes how extensions influence reflexivity in P-groups.

Findings

A precompact Abelian group of bounded order is reflexive iff its dual has no infinite compact subsets.

Extensions of reflexive P-groups are reflexive, but extensions involving compact or ω-bounded groups may not be.

P-modification of a reflexive σ-compact group can be nonreflexive.

Abstract

We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that: (1) A precompact Abelian group G of bounded order is reflexive iff the dual group $\hat{G}$ has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G. (2) Any extension of a reflexive P-group by another reflexive P-group is again reflexive. We show on the other hand that an extension of a compact group by a reflexive $\omega$-bounded group (even dual to a reflexive P-group) can fail to be reflexive. We also show that the P-modification of a reflexive $\sigma$-compact group can be nonreflexive (even if the P-modification of a locally compact Abelian group is always reflexive).