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A survey on strong reflexivity of abelian topological groups


Martín Peinador, Elena y Chasco, M.J. (2012) A survey on strong reflexivity of abelian topological groups. Topology and its Applications . ISSN 0166-8641 (Presentado)

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The Pontryagin duality Theorem for locally compact abelian groups (briey, LCA-groups)
has been the starting point for many different routes of research in Mathematics. From its appearance there was a big interest to obtain a similar result in a context broader than LCA-groups. Kaplan in the 40's proposed -and it remains open- the characterization of all the abelian topological groups for which the canonical mapping into its bidual is a topological isomorphism, assuming that the dual and the bidual carry the compact-open topology. Such groups are called reflexive.
In this survey we deal with results on reflexivity of certain classes of groups, with special emphasis on the class which better reflects the properties of LCA-groups, namely that of strongly reflexive groups. A topological abelian group is said to be strongly reflexive if all its closed subgroups and its Hausdorff quotients as well as the closed subgroups and the Hausdorff quotients of its dual group are reflexive.
By no means we can claim completeness of the survey: just an ordered view of the topic, with some small new results indicated in the text

Tipo de documento:Artículo
Palabras clave:Pontryagin duality theorem, Dual group, Reflexive group, Strongly reflexive group, Metrizable group, Čech-complete group, ω-bounded group, P-group
Materias:Ciencias > Matemáticas > Topología
Código ID:15139

S. Ardanza-Trevijano, M. J. Chasco, X. Dominguez and M. Tkachenko, Precompact noncompact reexive abelian grups, Forum Math. Vol 24 (2012) 289-302.

R. Arens, Duality in linear spaces, Duke Math J. 14 (1947) 787-794.

A. V. Arhangel'skii and M. G.Tkachenko, Topological Groups and Related Structures., Atlantis Press/World Scienti_c, Amsterdam-Paris, 2008.

L. Auβenhofer, Contributions to the Duality Theory of Abelian Topological Groups and to the Theory of Nuclear Groups, Dissertationes Mathematicae, CCCLXXXIV, Warszawa,


L. Auβenhofer, A duality property of an uncountable product of Z, Math. Z. 257 no 2 (2007) 231-237

L. Auβenhofer, A survey on nuclear groups , Nuclear Groups and Lie Groups, Research and Exposition in Mathematics, Volume 24. (Edited by E. Martín Peinador and J. Núñez

García) Heldermann Verlag, 2001

L. Auβenhofer, On the nuclearity of dual groups, Preprint, Passau.

L. Auβenhofer, D. Dikranjan and E. Martín Peinador, Locally quasi-convex compatible topologies on a topological group, Preprint 2010

W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Mathematics 1466. Springer-Verlag, Berlin Heidelberg, New York, 1991.

M. Banaszczyk and W. Banaszczyk, Characterization of nuclear spaces by means of additive subgroups. Math. Z. 186 (1984), 125-133.

W. Banaszczyk, Countable products of LCA groups: their closed subgroups, quotients and duality properties, Colloq. Math. 59 (1990), 52-57.

W. Banaszczyk, Pontryagin duality for subgroups and quotients of nuclear spaces, Math. Ann. 273 (1986), 653-664.

W. Banaszczyk, M. J. Chasco and E. Martín Peinador, Open subgroups and Pontryagin duality, Math. Z. (2) 215 (1994) 195-204.

W. Banaszczyk and E. Martín Peinador, The Glicksberg Theorem on Weakly Compact Sets for Nuclear Groups, Ann. N. Y. Acad. Sci., General Topology and Applications, Volume

788, no 1, (1996) pages 34-39.

R. Beattie, and H. P. Butzmann, Convergence structures and applications to functional analysis, (Kluwer Academic 2002).

R. Brown, P. J. Higgings and S. A. Morris, Countable products and sums of lines and circles:their closed subgroups, quotients and duality properties, Math. Proc. Cambridge Philos. Soc. 78 (1975), 19{32.

M. Bruguera, Grupos topológicos y grupos de convergencia: estudio de la dualidad de Pontryagin, Doctoral Dissertation. Barcelona, 1999.

M. Bruguera and M.J. Chasco, Strong reexivity of abelian groups, Czechoslovak Math. J. 51(126) (2001), no. 1, 213-224.

M. Bruguera and E. Martín Peinador. Banach-Dieudonné Theorem revisited, J. Aust. Math. Soc. 75 (2003), 69-83.

M. Bruguera and E. Martín Peinador, Open subgroups, compact subgroups and Binz-Butzmann reexivity, Topology Appl., 72 (1996) 101-111.

M. Bruguera, E. Martín Peinador and V. Tarieladze, Eberlein-Smulyan theorem for abelian topological groups, J. London Math. Soc., 70 (2) (2004) 341- 355

M. Bruguera, M.J. Chasco, E. Martín Peinador and V. Tarieladze, Completeness properties of locally quasi-convex groups, Topology Appl., 111, (2001) 81- 9.

M. Bruguera and M. Tkachenko, Duality in the class of precompact Abelian groups and the Baire property. To appear in Journal of Pure and Applied Algebra.

H.-P. Butzmann, Duality theory for convergence groups, Topology Appl. 111 (2001) 95-104.

M. J. Chasco, Pontryagin duality for metrizable groups. Arch. Math. 70 (1998), 22-28.

M.J. Chasco and X. Domínguez, Topologies on the direct sum of topological abelian groups,Topology Appl. 133 , no 3 (2003) 209-223.

M. J. Chasco and E. Martín Peinador, An approach to duality on Abelian precompact groups,J. Group Theory 11 (2008), 635-643.

M.J. Chasco and E. Martín Peinador, Binz-Butzmann duality versus Pontryagin Duality, Arch. Math. (Basel) 63 (3) (1994), 264-270.

M.J. Chasco and E. Martín Peinador, Pontryagin reexive groups are not determined by

their continuous characters, Rocky Mountain J. Math. 28 , no 1 (1998) 155-160.

M. J. Chasco, and E. Martín Peinador, On strongly reexive topological groups, Appl Gen Topol., 2(2) (2001), 219-226.

M. J. Chasco, E. Martín Peinador and V. Tarieladze, On Mackey topology for groups, Studia Math. 132 (3) (1999) 257-284

H. Chu, Compactifcation and duality of topological groups, Trans. Amer. Math. Soc. 123 (1966), 310-324.

W.W. Comfort, S. Hernáandez, D. Remus and F.J. Trigos-Arrieta, Open questions on topological groups, Nuclear Groups and Lie Groups, Research and Exposition in Mathematics, Volume 24. (Edited by E. Martín Peinador and J. Núñez García) Heldermann Verlag, 2001

W. W. Comfort, S. U. Raczkowski and F. J. Trigos Arrieta, The dual group of a dense subgroup, Czechoslovak Math. J. 54 (129) (2004) 509-533.

W.W. Comfort, S.U. Raczkowski and F.J. Trigos-Arrieta, Making group topologies with,and without, convergent sequences, Appl. Gen. Topology 7, (2006), no. 1.

W. W. Comfort and K. A. Ross, Topologies induced by groups of characters. Fund. Math. 55 (1964) 283-291.

W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacifc J. Math. 16 (1966) 483-496.

J.M. Díaz Nieto, On refinements of ω-bounded group topologies. Preprint 2011.

D. Dikranjan, Iv. Prodanov and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure and Applied Mathematics, vol. 130, Marcel Dekker

Inc., New York-Basel, 1989.

D. Dikanjan, E. Martín Peinador and V. Tarieladze, A class of metrizable locally quasiconvex groups which are not Mackey,

D. Dikranjan and M. Tkachenko, Sequential completeness of quotient groups, Bull. Austral. Math. Soc. 61 (2000) 129-151.

D. Dikranjan and D. Shakhmatov, Selected topics from the structure theory of topological groups, In Elliott Pearl, editor, Open problems in topology, 389{406. Elsevier, 2007.

D. Dikranjan and D. Shakhmatov, Quasi-convex density and determining subgroups of compact

abelian groups, J. Math. Anal. Appl. 362 (2010), 42-48

S. Gabriyelyan, Groups of quasi-invariance and the Pontryagin duality, Topology Appl. 157, no 18, (2010) 2786-2802.

S. Gabriyelyan, Reexive group topologies on Abelian groups, J. Group Theory 13, no. 6, (2010) 891{901.

J. Galindo, S. Hernández, and T. S. Wu, Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups, Open problems in topology II (E. Pearl, ed.), Elsevier Science. 2007

J. Galindo and S. Hernández, Pontryajin van-Kampen reexivity for free abelian topological groups, Forum Math. 11 (1999) 399-415.

J. Galindo and S. Macario, Pseudocompact group topologies with no infinite compact subsets, J. Pure and Appl. Algebra 215 (2011) 655-663.

J. Galindo, L. Recoder-Núñez and M. Tkachenko, Nondiscrete P-groups can be reexive, Topology Appl. 158 (2011) 194-203.

J. Galindo, L. Recoder-Núñez and M. Tkachenko, Reexivity of prodiscrete groups, J. Math. Anal. Appl. 384, no. 2 (2011) 320-330.

H. Glöckner, R. Gramlich and T. Hartnick, Final group topologies, Kac-Moody groups and Pontryagin duality, Israel J. Math. 177 (2010), 49-101

S. Hernández, Pontryagin duality for topological abelian groups, Math. Z. 238, no. 3, (2001) 493-503.

S. Hernáandez, Some new results about the Chu topology of discrete groups, Monats. Math., 149 (2006), 215-232

S. Hernández, The Bohr topology of discrete non-abelian groups, J. Lie Theory 18 (2008), no. 3, 733-746,

S. Hernández, S. Macario, F. J. Trigos-Arrieta, Uncountable products of determined groups need not be determined, J. Math. Anal. Appl. 348 (2008) 834-842.

S. Hern_andez and S. Macario, Dual properties in totally bounded Abelian groups, Arch. Math. 80 (2003) 271-283.

S. Hern_andez and J. Trigos, Group duality with the topology of precompact convergence J. Math. Anal. Appl, 303 (2005), 274-287.

S. Hernández and V. Uspenskij, Pontryagin Duality for Spaces of Continuous Functions, J. Math. Anal. Appl 242, no 2, (2000) 135-144

E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc, 64 (1948) 45-99.

E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Bd. 115, Academic Press, New York and Springer-Verlag, Berlin, 1963.

K. H. Hofmann and S. A. Morris, The structure of compact groups. De Gruyter Studies in Mathematics 25, 1998.

S. Kaplan, Extensions of the Pontryagin duality I: Infinite products, Duke Math. J. 15 (1948), 649-658.

S. Kaplan, Extensions of the Pontryagin duality II: Direct and inverse limits, Duke Math. J. 17 (1950), 419-435.

S. H. Kye, Pontryagin duality in real linear topological spaces, Chinese J. of Math., 12 (2) (1984) 129-136.

S.H. Kye, Several reexivities in topological vector spaces, J. Math. Anal. Appl. 139 (1989), no. 2, 477{482.

L. de Leo, Weak and strong topologies in topological abelian groups, PhD Thesis, Universidad Complutense de Madrid, July 2008.

H. Leptin, Zur Dualitatstheorie Projektiver Limites Abelscher Gruppen, Abh. Math. Sem. Univ. Hamburg 19 (1955) 264-268. MR 16, 899

E. Martín Peinador, A reflexive admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123 no. 11 (1995) 3563-3566.

E. Martín Peinador and V. Tarieladze, A property of Dunford-Pettis type in topological groups, Proc. Amer. Math. Soc., 132, (2004) 1827-1837

P. Nickolas, Reexivity of topological groups, Proc. Amer. Math. Soc. 65 (1977), 137-141

N. Noble, k-groups and duality , Trans. Amer. Math. Soc. 151 (1970), 551-561.

V. Pestov, Free Abelian topological groups and the Pontryagin-Van Kampen duality, Bull. Austr. Math. Soc., 52, (1995) 297-311

L. Pontrjagin, Topological groups. Princeton University Press, Princeton 1946. (Translated

from the Russian).

S. U. Raczkowski and J. Trigos. Duality of totally bounded Abelian groups, Bol. Soc. Mat. Mexicana, 3 (7) (2001) 1-12.

W. Roelcke, S. Dierolf, Uniform Structures on topological groups and their quotients, McGraw-Hill, New York, 1981.

M. F. Smith, The Pontrjagin duality theorem in linear spaces, Ann. of Math. 1952, 56 (2), 248-253.

M. G. Tkachenko, Compactness type properties in topological groups, Czechoslovak Math. J. 38 (113) (1988), 324-341

N. Th. Varopoulos, Studies in harmonic analysis, Proc. Camb. Philos. Soc. 60,(1964) 465-516

N.Ya. Vilenkin, The theory of characters of topological Abelian groups with boundedness given , Akad. Nauk SSSR., Izv. Math. 15 (1951), 439-462.

A. Weil, Sur les espaces à structure uniforme et sur la topologie générale, Publ. Math. Univ. Strasbourg, Hermann, Paris, 1937.

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