Bounded subgroups as a von Neumann radical of an Abelian group

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Abstract

For an infinite Abelian group G, we give a complete description of those nonzero bounded subgroups of G which are the von Neumann radical for some Hausdorff group topology on G. If G is of infinite exponent, we prove that for every nonzero bounded subgroup H of G there exists a Hausdorff group topology τ on G such that H is the von Neumann radical of (G, τ). If G has finite exponent, we show that the following are equivalent: (i) there exists a Hausdorff group topology τ on G such that H is the von Neumann radical of (G, τ); (ii) G contains a subgroup of the form Z(exp(H))(ω). In particular, an infinite Abelian group of finite exponent admits a Hausdorff minimally almost periodic group topology if and only if all its leading Ulm-Kaplansky invariants are infinite.

Original languageEnglish
Pages (from-to)185-199
Number of pages15
JournalTopology and its Applications
Volume178
DOIs
StatePublished - 1 Dec 2014

Keywords

  • Abelian group
  • Bounded Abelian group
  • Dual group
  • T-sequence
  • Von Neumann radical

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