Topological properties of the group of the null sequences valued in an Abelian topological group

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Abstract

For a Hausdorff Abelian topological group X, we denote by F0(X) the group of all X-valued null sequences endowed with the uniform topology. We prove that if X is an (E)-space (respectively, a strictly angelic space or a Š-space), then so is F0(X). We essentially simplify and clarify the theory of properties respected by the Bohr functor on Abelian topological groups, denoted below by X→X+. We prove that for a complete maximally almost periodic group X, the group X shares with X+ the same functionally bounded sets iff it shares the same compact sets and X+ is a μ-space. We show that for a locally compact Abelian (LCA) group X the following are equivalent: 1) X is totally disconnected, 2) F0(X) is a Schwartz group, 3) F0(X) respects compactness, 4) F0(X) has the Schur property. So, if a LCA group X is not totally disconnected, the group F0(X) is a reflexive non-Schwartz group which does not have the Schur property. We prove also that for every compact connected metrizable Abelian group X the group F0(X) is monothetic and every real-valued uniformly continuous function on F0(X) is bounded.

Original languageEnglish
Pages (from-to)136-155
Number of pages20
JournalTopology and its Applications
Volume207
DOIs
StatePublished - 1 Jul 2016

Keywords

  • (E)-space
  • Glicksberg property
  • Group of null sequences
  • Locally compact group
  • Monothetic group
  • Schur property
  • Strictly angelic space
  • Š-space

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