Precompact noncompact reflexive abelian groups.

*(English)*Zbl 1259.22001The duality properties of precompact Abelian groups have turned out to be more complicated than expected. This refers to the usual Pontryagin duality established between a topological group \(G\) and the Abelian group of its continuous characters \(G^\wedge\) (a character is understood as a homomorphism into the multiplicative group \(\mathbb{T}\) of complex numbers of modulus 1). This latter group equipped with the topology of uniform convergence on compact subsets is made into a topological group often denoted as \(G^\wedge\). The topological group \(G\) is then said to be reflexive when the evaluation mapping \(\alpha_{\text{ G}}: G \to G^{\wedge\wedge}\) is a topological isomorphism.

Reflexivity of topological groups has often been regarded as a sort of completeness property. Among the best known reflexive groups we can count locally compact groups (this is the classical Pontryagin-van Kampen duality theorem, to be found in most monographs in abstract harmonic analysis), additive groups of completely metrizable locally convex vector spaces (this can be deduced from an old paper of M. Smith [Ann. Math. (2) 56, 248–253 (1952; Zbl 0047.10701)], free Abelian topological groups on metrizable punctiform spaces as proved by S. Hernández and the reviewer [Forum Math. 11, No. 4, 399–415 (1999; Zbl 0924.22001)] or completely metrizable nuclear groups [W. Banaszczyk, Additive subgroups of topological vector spaces. Berlin etc.: Springer-Verlag (1991; Zbl 0743.46002)]. Even though examples of noncomplete reflexive groups are known, results as that of M. J. Chasco [Arch. Math. 70, No. 1, 22–28 (1998; Zbl 0899.22001)] who proved that a metrizable reflexive group must be complete or H. Leptin [Abh. Math. Sem. Univ. Hamburg 19, 264–268 (1955; Zbl 0065.01502)] who worked out the dual of a special countably compact group to see that it was discrete, led to wonder whether precompact reflexive groups should be compact.

The present paper answers that question by showing that any topological group in the class \(\mathcal{P}_h\) of pseudocompact groups whose countable subgroups have no discontinuous characters must be reflexive. Such pseudocompact groups had already been constructed by the last author of this paper back [Czech. Math. J. 38(113), 324–341 (1988; Zbl 0664.54006)].

Examples of noncompact pseudocompact reflexive groups of the same sort have been obtained independently by S. Macario and the reviewer [J. Pure Appl. Algebra 215, No. 4, 655–663 (2011; Zbl 1215.54015)] who also showed that most Abelian groups admitting a pseudocompact group topology (all of them if the singular cardinals hypothesis is assumed) can be made into \(\mathcal{P}_h\)-groups by a suitable choice of the topology.

The authors proceed with an analysis of the permanence properties of the class \(\mathcal{P}_h\) and show that every pseudocompact group can be obtained as a quotient of a group in \(\mathcal{P}_h\). Examples of precompact nonpseudocompact reflexive groups are obtained as well.

Since the appearance of the paper under review, the class of precompact reflexive groups has been considerably enriched by even countably compact examples, see, for instance, the papers by M. Tkachenko et al. [Topology Appl. 158, No. 2, 194–203 (2011; Zbl 1208.22002); J. Math. Anal. Appl. 384, No. 2, 320–330 (2011; Zbl 1228.22001)], M. Bruguera and M. Tkachenko [J. Pure Appl. Algebra 216, No. 12, 2636–2647 (2012; Zbl 1278.43003)], and the recent survey paper by M. J. Chasco et al. [Topology Appl. 159, No. 9, 2290–2309 (2012; Zbl 1247.22001)].

Reflexivity of topological groups has often been regarded as a sort of completeness property. Among the best known reflexive groups we can count locally compact groups (this is the classical Pontryagin-van Kampen duality theorem, to be found in most monographs in abstract harmonic analysis), additive groups of completely metrizable locally convex vector spaces (this can be deduced from an old paper of M. Smith [Ann. Math. (2) 56, 248–253 (1952; Zbl 0047.10701)], free Abelian topological groups on metrizable punctiform spaces as proved by S. Hernández and the reviewer [Forum Math. 11, No. 4, 399–415 (1999; Zbl 0924.22001)] or completely metrizable nuclear groups [W. Banaszczyk, Additive subgroups of topological vector spaces. Berlin etc.: Springer-Verlag (1991; Zbl 0743.46002)]. Even though examples of noncomplete reflexive groups are known, results as that of M. J. Chasco [Arch. Math. 70, No. 1, 22–28 (1998; Zbl 0899.22001)] who proved that a metrizable reflexive group must be complete or H. Leptin [Abh. Math. Sem. Univ. Hamburg 19, 264–268 (1955; Zbl 0065.01502)] who worked out the dual of a special countably compact group to see that it was discrete, led to wonder whether precompact reflexive groups should be compact.

The present paper answers that question by showing that any topological group in the class \(\mathcal{P}_h\) of pseudocompact groups whose countable subgroups have no discontinuous characters must be reflexive. Such pseudocompact groups had already been constructed by the last author of this paper back [Czech. Math. J. 38(113), 324–341 (1988; Zbl 0664.54006)].

Examples of noncompact pseudocompact reflexive groups of the same sort have been obtained independently by S. Macario and the reviewer [J. Pure Appl. Algebra 215, No. 4, 655–663 (2011; Zbl 1215.54015)] who also showed that most Abelian groups admitting a pseudocompact group topology (all of them if the singular cardinals hypothesis is assumed) can be made into \(\mathcal{P}_h\)-groups by a suitable choice of the topology.

The authors proceed with an analysis of the permanence properties of the class \(\mathcal{P}_h\) and show that every pseudocompact group can be obtained as a quotient of a group in \(\mathcal{P}_h\). Examples of precompact nonpseudocompact reflexive groups are obtained as well.

Since the appearance of the paper under review, the class of precompact reflexive groups has been considerably enriched by even countably compact examples, see, for instance, the papers by M. Tkachenko et al. [Topology Appl. 158, No. 2, 194–203 (2011; Zbl 1208.22002); J. Math. Anal. Appl. 384, No. 2, 320–330 (2011; Zbl 1228.22001)], M. Bruguera and M. Tkachenko [J. Pure Appl. Algebra 216, No. 12, 2636–2647 (2012; Zbl 1278.43003)], and the recent survey paper by M. J. Chasco et al. [Topology Appl. 159, No. 9, 2290–2309 (2012; Zbl 1247.22001)].

Reviewer: Jorge Galindo (Castellón)

##### MSC:

22A05 | Structure of general topological groups |

54H11 | Topological groups (topological aspects) |

22A10 | Analysis on general topological groups |

22B05 | General properties and structure of LCA groups |

43A40 | Character groups and dual objects |

##### Keywords:

precompact group; pseudocompact group; \(h\)-embedded subgroup; reflexive group; countably compact group; \(\mathcal{P}_h\)
PDF
BibTeX
XML
Cite

\textit{S. Ardanza-Trevijano} et al., Forum Math. 24, No. 2, 289--302 (2012; Zbl 1259.22001)

Full Text:
DOI

##### References:

[1] | Außenhofer L., Dissertationes Mathematicae pp 384– (1999) |

[2] | DOI: 10.1007/BF02571709 · Zbl 0790.22001 · doi:10.1007/BF02571709 |

[3] | DOI: 10.1007/s000130050160 · Zbl 0899.22001 · doi:10.1007/s000130050160 |

[4] | DOI: 10.1515/JGT.2008.039 · Zbl 1203.22002 · doi:10.1515/JGT.2008.039 |

[5] | Comfort W. W., Fund. Math. 55 pp 283– (1964) |

[6] | Comfort W. W., Pacific J. Math. 16 pp 483– (1966) |

[7] | DOI: 10.1016/j.topol.2004.07.012 · Zbl 1068.22001 · doi:10.1016/j.topol.2004.07.012 |

[8] | DOI: 10.1017/S0004972700022085 · Zbl 0943.22001 · doi:10.1017/S0004972700022085 |

[9] | DOI: 10.1515/form.2003.041 · Zbl 1058.22001 · doi:10.1515/form.2003.041 |

[10] | Glicksberg I., Can. J. Math. 14 pp 269– (1962) · Zbl 0109.02001 · doi:10.4153/CJM-1962-017-3 |

[11] | Grant D. L., Pacific J. Math. 68 pp 411– (1977) |

[12] | Hernández S., Archiv der Mathematik 80 pp 271– (2003) |

[13] | DOI: 10.1215/S0012-7094-48-01557-9 · Zbl 0034.30601 · doi:10.1215/S0012-7094-48-01557-9 |

[14] | DOI: 10.1007/BF02988875 · Zbl 0065.01501 · doi:10.1007/BF02988875 |

[15] | Raczkowski S. U., Bol. Soc. Mat. Mex. III 7 pp 1– (2001) |

[16] | Tkachenko M. G., Czech. Math. J. 38 pp 324– (1988) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.