The concept of boundedness and the Bohr compactification of a MAP Abelian group

Jorge Galindo; Salvador Hernández

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 3, page 195-218
  • ISSN: 0016-2736

Abstract

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Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an ℒ ∞ -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This result generalizes an analogous theorem of van Douwen for the discrete case and some other ones due to Hartman and Ryll-Nardzewski concerning the existence of I 0 -sets.  We also obtain some results on preservation of compactness for the Bohr topology of several types of MAP Abelian groups, like ℒ ∞ -groups, locally convex vector spaces and free Abelian topological groups.

How to cite

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Galindo, Jorge, and Hernández, Salvador. "The concept of boundedness and the Bohr compactification of a MAP Abelian group." Fundamenta Mathematicae 159.3 (1999): 195-218. <http://eudml.org/doc/212329>.

@article{Galindo1999,
abstract = {Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an $ℒ_∞$-group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This result generalizes an analogous theorem of van Douwen for the discrete case and some other ones due to Hartman and Ryll-Nardzewski concerning the existence of $I_0$-sets.  We also obtain some results on preservation of compactness for the Bohr topology of several types of MAP Abelian groups, like $ℒ_∞$-groups, locally convex vector spaces and free Abelian topological groups.},
author = {Galindo, Jorge, Hernández, Salvador},
journal = {Fundamenta Mathematicae},
keywords = {Bohr topology; LCA group; $ℒ_∞$-group; boundedness; locally convex vector space; DF-space; maximally almost periodic; respects compactness; C-embedded; C*-embedded; -groups; Abelian topological group; locally compact Abelian group; Montel spaces; locally convex reflexive real spaces; nuclear groups; nuclear locally convex spaces},
language = {eng},
number = {3},
pages = {195-218},
title = {The concept of boundedness and the Bohr compactification of a MAP Abelian group},
url = {http://eudml.org/doc/212329},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Galindo, Jorge
AU - Hernández, Salvador
TI - The concept of boundedness and the Bohr compactification of a MAP Abelian group
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 3
SP - 195
EP - 218
AB - Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an $ℒ_∞$-group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This result generalizes an analogous theorem of van Douwen for the discrete case and some other ones due to Hartman and Ryll-Nardzewski concerning the existence of $I_0$-sets.  We also obtain some results on preservation of compactness for the Bohr topology of several types of MAP Abelian groups, like $ℒ_∞$-groups, locally convex vector spaces and free Abelian topological groups.
LA - eng
KW - Bohr topology; LCA group; $ℒ_∞$-group; boundedness; locally convex vector space; DF-space; maximally almost periodic; respects compactness; C-embedded; C*-embedded; -groups; Abelian topological group; locally compact Abelian group; Montel spaces; locally convex reflexive real spaces; nuclear groups; nuclear locally convex spaces
UR - http://eudml.org/doc/212329
ER -

References

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