2010/12/28 by Dikran Dikranjan, Dikranjan, Dikran, Elena Martín Peinador +3 · 1 citation
Mathematics · #54C40 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #math.GN #msc:54C40
paper · pdf · doi:10.48550/arxiv.1012.5713
20 pages
arxiv created 2010/12/28 · openalex publication_date 2010/12/28 · arxiv updated 2010/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A topological group (G,μ) from a class \mathcal G of MAP topological abelian groups will be called a \it Mackey group in \mathcal G if it has the following property: if ν is a group topology in G such that (G,ν)∈ \mathcal G and (G,ν) has the same continuous characters, say (G,ν)\wedge=(G,μ)\wedge, then ν≤ μ. If \rmLCS is the class of Hausdorff topological abelian groups which admit a structure of a locally convex topological vector space over \mathbb R, it is well-known that every metrizable (G,μ) ∈ \rmLCS is a Mackey group in \rmLCS. For the class \rmLQC of locally quasi-convex Hausdorff topological abelian groups, it was proved in 1999 that every \bf complete metrizable (G,μ)∈ \rmLQC is a Mackey group in \rmLQC (\citeCMPT). The completeness cannot be \NB dropped within the class \rmLQC as we prove in this paper. In fact, we provide a large family of metrizable precompact \NB(noncompact) groups which \bf are not Mackey groups in \rmLQC (Theorem \refbasth). Those examples are constructed from groups of the form c0(X), whose elements are the null sequences of a topological abelian group X, and whose topology is the uniform topology. We first show that for a compact metrizable group X≠\0\ the topological group c0(X) is a non-compact complete metrizable locally quasi-convex group, which has \bf countable topological dual iff X is connected. Then we prove that for a connected compact metrizable group X≠\0\ the group c0(X) endowed with the product topology induced from the product X\N is metrizable precompact but not a Mackey group in LQC.