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Precompact noncompact reflexive abelian groups. (English) Zbl 1259.22001
The 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)].

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
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