RT Journal Article
T1 Locally Quasi-Convex Compatible Topologies on a Topological Group
A1 Außenhofer, Lydia
A1 Dikranjan, Dikran
A1 Martín Peinador, Elena
AB For a locally quasi-convex topological abelian group (G, τ), we study the poset C (G, τ) of all locally quasi-convex topologies on G that are compatible with τ (i.e., have the same dual as (G, τ)) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G, Gb). Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates “from below”, our strategy consists of finding appropriate subgroups H of G that are easier to handle and show that C (H) and C (G/H) are large and embed, as a poset, in C (G, τ). Important special results are: (i) if K is a compact subgroup of a locally quasi-convex group G, then C (G) and C (G/K) are quasi-isomorphic (3.15); (ii) if D is a discrete abelian group of infinite rank, then C (D) is quasi-isomorphic to the poset FD of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group G with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset C (G) is as big as the underlying topological structure of (G, τ) (and set theory) allows. For a metrizable connected compact group X, the group of null sequences G = c0(X) with the topology of uniform convergence is studied. We prove that C (G) is quasi-isomorphic to P(R) (6.9).
PB MDPI
SN 2075-1680
YR 2015
FD 2015
LK https://hdl.handle.net/20.500.14352/34254
UL https://hdl.handle.net/20.500.14352/34254
LA eng
NO 1. Varopoulos, N.T. Studies in harmonic analysis. Proc. Camb. Phil. Soc. 1964, 60, 467–516.2. Chasco, M.J.; Martín-Peinador, E.; Tarieladze, V. On Mackey topology for groups. Stud. Math.1999, 132, 257–284.3. De Leo, L. Weak and Strong Topologies in Topological Abelian Groups. Ph.D. Thesis, UniversidadComplutense de Madrid, Madrid, Spain, July 2008.4. De Leo, L.; Dikranjan, D.; Martín-Peinador, E.; Tarieladze, V. Duality Theory for Groups Revisited:g-barrelled groups, Mackey & Arens Groups. 2015, in preparation.5. Bonales, G.; Trigos-Arrieta, F.J.; Mendoza, R.V. A Mackey-Arens theorem for topological Abeliangroups. Bol. Soc. Mat. Mex. III 2003, 9, 79–88.6. Berarducci, A.; Dikranjan, D.; Forti, M.; Watson, S. Cardinal invariants and independence resultsin the lattice of precompact group topologies. J. Pure Appl. 1998, 126, 19–49.7. Comfort, W.; Remus, D. Long chains of Hausdorff topological group topologies. J. Pure Appl.Algebra 1991, 70, 53–72.8. Comfort, W.; Remus, D. Long chains of topological group topologies—A continuation. TopologyAppl. 1997, 75, 51–79.9. Dikranjan, D. The Lattice of Compact Representations of an infinite group. In Proceedings ofGroups 93, Galway/St Andrews Conference, London Math. Soc. Lecture Notes 211; CambidgeUniv. Press: Cambridge, UK, 1995; pp. 138–155.10. Dikranjan, D. On the poset of precompact group topologies. In Topology with Applications,Proceedings of the 1993 Szekszàrd (Hungary) Conference, Bolyai Society Mathematical Studies;Czászár, Á., Ed.; Elsevier: Amsterdam, The Netherlands, 1995; Volume 4, pp. 135–149.11. Dikranjan, D. Chains of pseudocompact group topologies. J. Pure Appl. Algebra 1998, 124,65–100.12. Engelking, R. General Topology, (Sigma Series in Pure Mathematics, 6), 2nd ed.; HeldermannVerlag: Berlin, Germany, 1989.13. Abramsky, S.; Jung, A. Domain theory. In Handbook of Logic in Computer Science III; Abramsky,S., Gabbay, D.M., Maibaum, T.S.E., Eds.; Oxford University Press: New York, NY, USA, 1994;pp. 1–168.14. Banaszczyk, W. Additive Subgroups of Topological Vector Spaces, Lecture Notes in Mathematics;Springer Verlag: Berlin, Germany, 1991; Volume 1466.15. Enflo, P. Uniform structures and square roots in topological groups. Israel J. Math. 1970, 8,230–252.16. Außenhofer, L.; Dikranjan, D.; Martín-Peinador, E. Locally quasi-convex compatible topologieson σ-compact LCA groups. 2015, in preparation.17. Fuchs, L. Infinite Abelian Groups; Academic Press: New York, NY, USA, 1970.18. Dikranjan, D.; Shakhmatov, D. Topological groups with many small subgroups. Topology Appl.2015, in press.19. Dikranjan, D.; Prodanov, I.; Stojanov, L. Topological Groups (Characters, Dualities, and MinimalGroup Topologies); Marcel Dekker, Inc.: New York, NY, USA, 1990.20. Bruguera, M.; Martín-Peinador, E. Open subgroups, compact subgroups and Binz-Butzmannreflexivity. Topology Appl. 1996, 72, 101–111.21. Außenhofer, L. A note on weakly compact subgroups of locally quasi-convex groups. Arch. Math.2013, 101, 531–540.22. Dikranjan, D.; Protasov, I. Counting maximal topologies on countable groups and rings. TopologyAppl. 2008, 156, 322–325.23. Dikranjan, D.; Martín-Peinador, E.; Tarieladze, V. Group valued null sequences and metrizablenon-Mackey groups. Forum Math. 2014, 26, 723–757.24. Sierpinski, W. Cardinal and ordinal numbers; Panstwowe Wydawnictwo Naukowe: Warsaw,Poland, 1958.25. Baumgartner, J.E. Almost disjoint sets, the dense set problem and the partition calculus. Ann.Math. Logic 1976, 10, 401–439.26. De la Barrera Mayoral, D.; Dikranjan, D.; Martìn Peinador, E. “Varopoulos paradigm": Mackeyproperty vs. metrizability in topological groups. 2015, in preparation.27. Außenhofer, L.; de la Barrera Mayoral, D. Linear topologies on Z are not Mackey topologies.J. Pure Appl. Algebra 2012, 216, 1340–1347.28. De la Barrera Mayoral, D. Q is not Mackey group. Topology Appl. 2014, 178, 265–275
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RD 14 may 2024