Groups of quasi-invariance and the Pontryagin duality

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29 Scopus citations


A Polish group G is called a group of quasi-invariance or a QI-group, if there exist a locally compact group X and a probability measure μ on X such that (1) there exists a continuous monomorphism ø from G into X with dense image, and (2) for each gεX either gεø(G) and the shift μg is equivalent to μ or g∉ø(G) and μg is orthogonal to μ. It is proved that ø(G) is a σ-compact subset of X. We show that there exists a Polish non-locally quasi-convex (and hence nonreflexive) QI-group such that its bidual is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of X. It is constructed a reflexive non-discrete group topology on the integers.

Original languageEnglish
Pages (from-to)2786-2802
Number of pages17
JournalTopology and its Applications
Issue number18
StatePublished - 1 Dec 2010


  • Dual group
  • Group of quasi-invariance
  • Polish group
  • Pontryagin duality theorem
  • Quasi-convex group
  • T-sequence

ASJC Scopus subject areas

  • Geometry and Topology


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