On characterized subgroups of Abelian topological groups X and the group of all X -valued null sequences

S. S. Gabriyelyan

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 1, page 73-99
  • ISSN: 0010-2628

Abstract

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Let X be an Abelian topological group. A subgroup H of X is characterized if there is a sequence 𝐮 = { u n } in the dual group of X such that H = { x ∈ X : ( u n , x ) → 1 } . We reduce the study of characterized subgroups of X to the study of characterized subgroups of compact metrizable Abelian groups. Let c 0 ( X ) be the group of all X -valued null sequences and 𝔲 0 be the uniform topology on c 0 ( X ) . If X is compact we prove that c 0 ( X ) is a characterized subgroup of X ℕ if and only if X ≅ 𝕋 n × F , where n ≥ 0 and F is a finite Abelian group. For every compact Abelian group X , the group c 0 ( X ) is a 𝔤 -closed subgroup of X ℕ . Some general properties of ( c 0 ( X ) , 𝔲 0 ) and its dual group are given. In particular, we describe compact subsets of ( c 0 ( X ) , 𝔲 0 ) .

How to cite

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Gabriyelyan, S. S.. "On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences." Commentationes Mathematicae Universitatis Carolinae 55.1 (2014): 73-99. <http://eudml.org/doc/260783>.

@article{Gabriyelyan2014,
abstract = {Let $X$ be an Abelian topological group. A subgroup $H$ of $X$ is characterized if there is a sequence $\mathbf \{u\} = \lbrace u_n\rbrace $ in the dual group of $X$ such that $H= \lbrace x\in X: \; (u_n,x)\rightarrow 1\rbrace $. We reduce the study of characterized subgroups of $X$ to the study of characterized subgroups of compact metrizable Abelian groups. Let $c_0(X)$ be the group of all $X$-valued null sequences and $\mathfrak \{u\}_0$ be the uniform topology on $c_0(X)$. If $X$ is compact we prove that $c_0(X)$ is a characterized subgroup of $X^\mathbb \{N\}$ if and only if $X\cong \mathbb \{T\}^n\times F$, where $n\ge 0$ and $F$ is a finite Abelian group. For every compact Abelian group $X$, the group $c_0(X)$ is a $\mathfrak \{g\}$-closed subgroup of $X^\mathbb \{N\}$. Some general properties of $(c_0(X),\mathfrak \{u\}_0)$ and its dual group are given. In particular, we describe compact subsets of $(c_0(X),\mathfrak \{u\}_0)$.},
author = {Gabriyelyan, S. S.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {group of null sequences; $T$-sequence; characterized subgroup; $T$-characterized subgroup; $\mathfrak \{g\}$-closed subgroup; group of null sequences; -sequence; characterized subgroup; -characterized subgroup; -closed subgroup},
language = {eng},
number = {1},
pages = {73-99},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences},
url = {http://eudml.org/doc/260783},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Gabriyelyan, S. S.
TI - On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 1
SP - 73
EP - 99
AB - Let $X$ be an Abelian topological group. A subgroup $H$ of $X$ is characterized if there is a sequence $\mathbf {u} = \lbrace u_n\rbrace $ in the dual group of $X$ such that $H= \lbrace x\in X: \; (u_n,x)\rightarrow 1\rbrace $. We reduce the study of characterized subgroups of $X$ to the study of characterized subgroups of compact metrizable Abelian groups. Let $c_0(X)$ be the group of all $X$-valued null sequences and $\mathfrak {u}_0$ be the uniform topology on $c_0(X)$. If $X$ is compact we prove that $c_0(X)$ is a characterized subgroup of $X^\mathbb {N}$ if and only if $X\cong \mathbb {T}^n\times F$, where $n\ge 0$ and $F$ is a finite Abelian group. For every compact Abelian group $X$, the group $c_0(X)$ is a $\mathfrak {g}$-closed subgroup of $X^\mathbb {N}$. Some general properties of $(c_0(X),\mathfrak {u}_0)$ and its dual group are given. In particular, we describe compact subsets of $(c_0(X),\mathfrak {u}_0)$.
LA - eng
KW - group of null sequences; $T$-sequence; characterized subgroup; $T$-characterized subgroup; $\mathfrak {g}$-closed subgroup; group of null sequences; -sequence; characterized subgroup; -characterized subgroup; -closed subgroup
UR - http://eudml.org/doc/260783
ER -

References

top
  1. Arhangel'skii A.V., Open and close-to-open mappings. Relations among spaces, Trudy Moskov. Mat. Obsch. 15 (1966), 181–223. MR0206909
  2. Arhangel'skii A.V., Tkachenko M.G., Topological Groups and Related Structures, Atlantis Press/World Scientific, Amsterdam-Paris, 2008. MR2433295
  3. Armacost D., The Structure of Locally Compact Abelian Groups, Monographs and Textbooks in Pure and Applied Mathematics, 68, Marcel Dekker, Inc., New York, 1981. Zbl0509.22003MR0637201
  4. Aussenhofer L., Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups, Dissertation, Tübingen, 1998; Dissertationes Math. (Rozprawy Mat.) 384 (1999), 113p. Zbl0953.22001MR1736984
  5. Barbieri G., Dikranjan D., Milan C., Weber H., 10.1016/S0166-8641(02)00366-8, Topology Appl. 132 (2003), 89–101. MR1990082DOI10.1016/S0166-8641(02)00366-8
  6. Barbieri G., Dikranjan D., Milan C., Weber H., Convergent sequences in precompact group topologies, Appl. Gen. Topol. 6 (2005), 149–169. Zbl1092.22001MR2195810
  7. Barbieri G., Dikranjan D., Milan C., Weber H., 10.1002/mana.200510650, Math. Nachr. 281 (2008), 930–950. Zbl1144.11059MR2431568DOI10.1002/mana.200510650
  8. Beiglböck M., Steineder C., Winkler R., Sequences and filters of characters characterizing subgroups of compact Abelian groups, Topology Appl. 153 (2006), 1682–1695. Zbl1091.22001MR2227021
  9. Biró A., 10.1016/j.jnt.2006.03.008, J. Number Theory 121 (2006), 324–354. Zbl1153.11031MR2274908DOI10.1016/j.jnt.2006.03.008
  10. Biró A., 10.5802/jtnb.603, J. Théor. Nombres Bordeaux 19 (2007), 567–582. Zbl1159.11022MR2388789DOI10.5802/jtnb.603
  11. Bíró A., Déshouillers J.-M., Sós V.T., Good approximation and characterization of subgroups of ℝ / ℤ , Studia Sci. Math. Hungar. 38 (2001), 97–113. MR1877772
  12. Biró A., Sós V.T., 10.1016/S0022-314X(02)00068-9, J. Number Theory 99 (2003), 405–414. Zbl1058.11047MR1968461DOI10.1016/S0022-314X(02)00068-9
  13. Borel J.P., 10.5802/afst.595, Ann. Fac. Sci. Toulouse Math. 5 (1983), 217–235. MR0747191DOI10.5802/afst.595
  14. Chasco M.J., 10.1007/s000130050160, Arch. Math. 70 (1998), 22–28. Zbl0899.22001MR1487450DOI10.1007/s000130050160
  15. Chasco M.J., Martín-Peinador E., Tarieladze V., On Mackey topology for groups, Studia Math. 132 (1999), 257–284. Zbl0930.46006MR1669662
  16. Comfort W.W., Raczkowski S.U., Trigos-Arrieta F.J., 10.4995/agt.2006.1936, Appl. Gen. Topol. 7 (2006), 109–124. Zbl1135.22004MR2284939DOI10.4995/agt.2006.1936
  17. Comfort W.W., Ross K.A., Topologies induced by groups of characters, Fund. Math. 55 (1964), 283–291. MR0169940
  18. Comfort W., Trigos-Arrieta F.-J., Ta Sun Wu, The Bohr compactification, modulo a metrizable subgroup, Fund. Math. 143 (1993), 119–136. Zbl0872.22001MR1240629
  19. Dikranjan D., Gabriyelyan S.S., 10.1016/j.topol.2013.07.037, Topology Appl. 160 (2013), 2427–2442. MR3120657DOI10.1016/j.topol.2013.07.037
  20. Dikranjan D., Kunen K., 10.1016/j.jpaa.2005.12.011, J. Pure Appl. Algebra 208 (2007), 285–291. Zbl1109.22002MR2269843DOI10.1016/j.jpaa.2005.12.011
  21. Dikranjan D., Martín-Peinador E., Tarieladze V., Group valued null sequences and metrizable non-Mackey groups, Forum Math. DOI: 10.1515/forum-2011-0099, February 2012. 
  22. Dikranjan D., Milan C., Tonolo A., 10.1016/j.jpaa.2004.08.021, J. Pure Appl. Algebra 197 (2005), 23–41. MR2123978DOI10.1016/j.jpaa.2004.08.021
  23. Dikranjan D., Prodanov I., Stojanov L., Topological groups. Characters, Dualities, and Minimal Group Topologies, Marcel Dekker, Inc., New York-Basel, 1990. MR1015288
  24. Eggleston H.G., Sets of fractional dimensions which occur in some problems of number theory, Proc. London Math. Soc. 54 (1952), 42–93. Zbl0045.16603MR0048504
  25. Engelking R., General Topology, Panstwowe Wydawnictwo Naukowe, Waszawa, 1977. Zbl0684.54001MR0500779
  26. Gabriyelyan S.S., 10.1016/j.topol.2010.08.018, Topology Appl. 157 (2010), 2786–2802. Zbl1207.54047MR2729338DOI10.1016/j.topol.2010.08.018
  27. Gabriyelyan S.S., 10.1016/j.topol.2010.09.002, Topology Appl. 157 (2010), 2834–2843. Zbl1221.22009MR2729343DOI10.1016/j.topol.2010.09.002
  28. Gabriyelyan S.S., 10.4064/fm221-2-1, Fund. Math. 221 (2013), 95–127. MR3062915DOI10.4064/fm221-2-1
  29. Gabriyelyan S.S., Minimally almost periodic group topologies on infinite countable Abelian groups: A solution to a question by Comfort, preprint, 2010, arXiv.org/abs/1002.1468. 
  30. Gabriyelyan S.S., On T -characterized subgroups of compact Abelian groups, preprint, 2012. 
  31. Glicksberg I., 10.4153/CJM-1962-017-3, Canad. J. Math. 14 (1962), 269–276. Zbl0109.02001MR0155923DOI10.4153/CJM-1962-017-3
  32. Graev M., Free topological groups, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 278–324. MR0025474
  33. Hart J.E., Kunen K, 10.1016/j.topol.2003.08.036, Topology Appl. 151 (2005), 157–168. Zbl1074.54013MR2139749DOI10.1016/j.topol.2003.08.036
  34. Hart J.E., Kunen K., 10.1016/j.topol.2005.02.011, Topology Appl. 153 (2006), 991–1002. Zbl1093.54010MR2203013DOI10.1016/j.topol.2005.02.011
  35. Hewitt E., Ross K.A., Abstract Harmonic Analysis, Vol. I, 2nd ed., Springer, Berlin, 1979. Zbl0837.43002MR0551496
  36. Kaplan S., 10.1215/S0012-7094-48-01557-9, Duke Math. J. 15 (1948), 649–658. MR0026999DOI10.1215/S0012-7094-48-01557-9
  37. Kraaikamp C., Liardet P., 10.1090/S0002-9939-1991-1062392-7, Proc. Amer. Math. Soc. 112 (1991), 303–309. Zbl0725.11033MR1062392DOI10.1090/S0002-9939-1991-1062392-7
  38. Larcher G., 10.1090/S0002-9939-1988-0947645-2, Proc. Amer. Math. Soc. 103 (1988), 718–722. Zbl0659.10009MR0947645DOI10.1090/S0002-9939-1988-0947645-2
  39. Nienhuys J., Construction of group topologies on Abelian groups, Fund. Math. 75 (1972), 101–116. Zbl0213.03503MR0302810
  40. Petersen K., Shapiro L., 10.1090/S0002-9947-1973-0322839-1, Trans. Amer. Math. Soc. 177 (1973), 375–390. Zbl0229.54036MR0322839DOI10.1090/S0002-9947-1973-0322839-1
  41. Protasov I.V., Zelenyuk E.G., 10.1070/IM1991v037n02ABEH002071, Math. USSR Izv. 37 (1991), 445–460; Russian original: Izv. Akad. Nauk SSSR 54 (1990), 1090–1107. Zbl0997.22002MR1086087DOI10.1070/IM1991v037n02ABEH002071
  42. Raczkowski S.U., 10.1016/S0166-8641(01)00110-9, Topology Appl. 121 (2002), 63–74. Zbl1007.22003MR1903683DOI10.1016/S0166-8641(01)00110-9
  43. Rolewicz S., Some remarks on monothetic groups, Colloq. Math. 13 (1964), 28–29. Zbl0131.26901MR0171876
  44. Siwiec F., 10.1016/0016-660X(71)90120-6, General Topology Appl. 1 (1971), 143–154. MR0288737DOI10.1016/0016-660X(71)90120-6
  45. Winkler R., 10.1007/s006050200027, Monatsh. Math. 135 (2002), 333–343. Zbl1008.37005MR1914810DOI10.1007/s006050200027

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