PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1999 | 159 | 3 | 195-218
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Twórcy
  • Departamento de Matemáticas, Universidad Jaume I, 12071 Castellón, Spain, jgalindo@mat.uji.es
  • Departamento de Matemáticas, Universidad Jaume I, 12071 Castellón, Spain, hernande@mat.uji.es
Bibliografia
  • [1] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin, 1991.
  • [2] W. Banaszczyk and E. Martín-Peinador, The Glicksberg Theorem on weakly compact sets for nuclear groups, Ann. New York Acad. Sci. 788 (1996), 34-39.
  • [3] W. W. Comfort, S. Hernández and F. J. Trigos-Arrieta, Relating a locally compact Abelian group to its Bohr compactification, Adv. Math. 120 (1996), 322-344.
  • [4] W. W. Comfort and K. A. Ross, Topologies induced by groups of characters, Fund. Math. 55 (1964), 283-291.
  • [5] W. W. Comfort, F. J. Trigos-Arrieta, and T.-S. Wu, The Bohr compactification, modulo a metrizable subgroup, Fund. Math. 143 (1993), 119-136.
  • [6] E. K. van Douwen, The maximal totally bounded group topology on G and the biggest minimal G-space, for Abelian groups G, Topology Appl. 34 (1990), 69-91.
  • [7] J. Galindo, Acotaciones y topologías débiles sobre grupos abelianos maximalmente casi periódicos, Ph.D. Thesis, Universidad Jaume I, Castellón, 1997.
  • [8] J. Galindo and S. Hernández, On a theorem of van Douwen, preprint, 1998.
  • [9] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, New York, 1960.
  • [10] I. Glicksberg, Uniform boundedness for groups, Canad. J. Math. 14 (1962), 269-276.
  • [11] S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, Colloq. Math. 12 (1964), 29-39.
  • [12] S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions II, ibid. 15 (1966), 79-86.
  • [13] J. Hejcman, Boundedness in uniform spaces and topological groups, Czechoslovak Math. J. 9 (1962), 544-562.
  • [14] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Grundlehren Math. Wiss. 115, Springer, Berlin, 1963.
  • [15] R. Hughes, Compactness in locally compact groups, Bull. Amer. Math. Soc. 79 (1973), 122-123.
  • [16] G. Köthe, Topological Vector Spaces, Vol. I, Grundlehren Math. Wiss. 159, Springer, Berlin, 1969.
  • [17] A. G. Leiderman, S. A. Morris and V. G. Pestov, The free abelian topological group and the free locally convex space on the unit interval, J. London Math. Soc. 56 (1997), 529-538.
  • [18] M. Moskowitz, Uniform boundedness for non-abelian groups, Math. Proc. Cambridge Philos. Soc. 97 (1985), 107-110.
  • [19] N. Noble, k-groups and duality, Trans. Amer. Math. Soc. 151 (1970), 551-561.
  • [20] V. Pestov, Free Abelian topological groups and the Pontryagin-van Kampen duality, Bull. Austral. Math. Soc. 52 (1995), 297-311.
  • [21] D. Remus and F. J. Trigos-Arrieta, Abelian groups which satisfy Pontryagin duality need not respect compactness, Proc. Amer. Math. Soc. 117 (1993), 1195-1200.
  • [22] C. Ryll-Nardzewski, Concerning almost periodic extensions of functions, Colloq. Math. 12 (1964), 235-237.
  • [23] L. J. Sulley, On countable inductive limits of locally compact abelian groups, J. London Math. Soc. (2) 5 (1972), 629-637.
  • [24] M. Tkachenko, Completeness of free Abelian topological groups, Dokl. Akad. Nauk SSSR 269 (1983), 299-303 (in Russian); English transl.: Soviet Math. Dokl. 27 (1983), 341-345.
  • [25] F. J. Trigos-Arrieta, Continuity, boundedness, connectedness and the Lindelöf property for topological groups, J. Pure Appl. Algebra 70 (1991), 199-210.
  • [26] V. V. Uspenskiĭ, On the topology of a free locally convex space, Dokl. Akad. Nauk SSSR 270 (1983), 1334-1337 (in Russian); English transl.: Soviet Math. Dokl. 27 (1983), 781-785.
  • [27] V. V. Uspenskiĭ, Free topological groups of metrizable spaces, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 1295-1319 (in Russian); English transl.: Math. USSR-Izv. 37 (1991), 657-680.
  • [28] M. Valdivia, The space of distributions D' is not $B_r$-complete, Math. Ann. 211 (1974), 145-149.
  • [29] M. Valdivia, The space D(Ω) is not $B_r$-complete, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 4, 29-43.
  • [30] N. Varopoulos, Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 (1964), 465-516.
  • [31] R. Venkataraman, Characterization, structure and analysis on abelian $ℒ_∞$ groups, Monatsh. Math. 100 (1985), 47-66.
  • [32] N. Ya. Vilenkin, Theory of characters of topological Abelian groups with a given boundedness, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 439-462 (in Russian).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv159i3p195bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.