Abstract
For a topological Abelian group X, we consider in the group X ℕ the uniform topology and study some properties of the obtained topological group. In particular, we show, that if \( X\kern0.5em =\kern0.5em \mathbb{S} \) is the circle group, then the group \( {\mathbb{S}}^{\mathbb{N}} \) endowed with the uniform topology has the dual group of cardinality 2c.
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References
A. V. Arhangel’skii and M. G. Tkachenko, Topological Groups and Related Structures, Atlantis Series in Math., Vol. 1, Atlantis Press, World Scientific, Paris–Amsterdam (2008).
H. Anzai and S. Kakutani, “Bohr compactifications of a locally compact Abelian group, II,” Proc. Imp. Acad. Tokyo, 19, 533–539 (1943).
L. Aussenhofer, “Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups,” Disser. Math., 384, Warsaw (1999).
W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lect. Notes Math., 1466, Springer-Verlag, Berlin (1991).
N. Bourbaki, Éléments de Mathématique. Topologie Générale. Chapitre X: Espaces Fonctionels, Hermann, Paris (1974).
M. J. Chasco, E. Martín-Peinador, and V. Tarieladze, “On Mackey topology for groups,” Stud. Math., 132, No. 3, 257–284 (1999).
W. W. Comfort and K. A. Ross, “Pseudocompactness and uniform continuity in topological groups,” Pac. J. Math., 16, 483–496 (1966).
D. Dikranjan, E. Martín-Peinador, and V. Tarieladze, “Group valued null sequences and metrizable non-Mackey groups,” in: Forum Math., published online 2012.02.03.
D. Dikranjan, Iv. Prodanov, and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure Appl. Math., 130, Marcel Dekker, New York–Basel (1989).
R. Engelking, General Topology, Panstwowe Wydawnictwo Naukowe, Warszawa (1985).
G. Fichtenhollz and L. Kantorovitch, “Sur les operations lineaires dans l’espace ses fonctions bornees,” Stud. Math., 5, 69–98 (1934).
S. S. Gabriyelyan, “Groups of quasi-invariance and the Pontryagin duality,” Topol. Appl., 157, 2786–2802 (2010).
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer-Verlag, Berlin (1963).
S. Kakutani, “On cardinal numbers related with a compact Abelian group,” Proc. Imp. Acad. Tokyo, 19, 366–372 (1943).
J. W. Nienhuys, “A solenoidal and monothetic minimally almost periodic group,” Fund. Math., 73, No. 2, 167–169 (1971/72).
S. Rolewicz, “Some remarks on monothetic groups,” Colloq. Math., 13, 28–29 (1964).
L. J. Sulley, “On countable inductive limits of locally compact Abelian groups,” J. London Math. Soc., 5, 629–637 (1972).
N. Ya. Vilenkin, “The theory of characters of topological Abelian groups with boundedness given,” Izv. Akad. Nauk SSSR, Ser. Mat., 15, 439–162 (1951).
A. Weil, L’intégration dans les groupes topologiques et ses applications, Hermann, Paris (1940).
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. Algebra and Topology, 91, 2014.
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Dikranjan, D., Martin-Peinador, E. & Tarieladze, V. Countable Powers of Compact Abelian Groups in the Uniform Topology and Cardinality of Their Dual Groups. J Math Sci 211, 127–135 (2015). https://doi.org/10.1007/s10958-015-2603-2
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DOI: https://doi.org/10.1007/s10958-015-2603-2