Publication: Group valued null sequences and metrizable non-Mackey groups
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2014-05
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WALTER DE GRUYTER
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Abstract
For a topological abelian group X we topologize the group c0(X) of all X-valued null sequences in a way such that when X= the topology of c0() coincides with the usual Banach space topology of the classical Banach space c0. If X is a non-trivial compact connected metrizable group, we prove that c0(X) is a non-compact Polish locally quasi-convex group with countable dual group c0(X). Surprisingly, for a compact metrizable X, countability of c0(X) leads to connectedness of X. Our principal application of the above results is to the class of locally quasi-convex Mackey groups (LQC-Mackey groups). A topological group (G,) from a class of topological abelian groups will be called a Mackey group in or a -Mackey group if it has the following property: if is a group topology in G such that (G,) and (G,) has the same character group as (G,), then . Based upon the results obtained for c0(X), we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC-Mackey. Namely, we show that for a connected compact metrizable group X0, the group c0(X), endowed with the topology induced from the product topology on X, is a metrizable precompact connected group which is not a Mackey group in LQC. Since metrizable locally convex spaces always carry the Mackey topology – a well-known fact from Functional Analysis –, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important role.
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H. Anzai and S. Kakutani, Bohr compactifications of a locally compact Abelian group. II, Proc. Imp. Acad. Tokyo 19 (1943), 533–539.
L. Aussenhofer, Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Diss. Math. CCCLXXXIV, Warsaw, 1999.
L. Aussenhofer and D. de la Barrera, Linear topologies on Z are not Mackey topologies, preprint (2011), http://arxiv.org/abs/1107.2661.
W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Mathematics 1466, Springer-Verlag, Berlin, 1991.
M. Barr and H. Kleisli, On Mackey topologies in topological abelian groups, Theory Appl. Categ. 8 (2001), 54–62
G. Bennett and N. J. Kalton, Inclusion theorems for K-spaces, Canad. J. Math. 25
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, Pacific J. Math. 16 (1966), 483–496.
L. De Leo, D. Dikranjan, E. Martín-Peinador and V. Tarieladze, Duality theory for groups revisited: g-barrelled, Mackey and Arens groups, preprint (2009).
D. Dikranjan, E. Martín-Peinador and V. Tarieladze, A class of metrizable locally quasi-convex groups which are not Mackey, preprint (2010), http://arxiv.org/abs/1012.5713.
D. Dikranjan, I. Prodanov and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure and Applied Mathematics 130, Marcel Dekker, New York, 1989.
D. Dikranjan and D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on abelian groups, Topology Appl. 151 (2005), no. 1–3, 2–54.
P. Enflo, Uniform structures and square roots in topological groups, Israel J. Math. 8 (1970), 230–252.
S. S. Gabriyelyan, Groups of quasi-invariance and the Pontryagin duality, Topology Appl. 157 (2010), no. 18, 2786–2802.
J. Galindo and S. Garcia-Ferreira, Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences, Topology Appl. 154 (2007), no. 2, 476–490.
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Die Grundlehren der Mathematischen Wissenschaften 115, Springer-Verlag, Berlin, 1963.
S. Kaplan, Extension from the Pontryagin duality I: Infinite products, Duke Math. J. 15 (1948), 649–658.
J. L. Kelley and I. Namioka, Linear Topological Spaces, Springer-Verlag, New York, 1963.
J.W. Nienhuys, A solenoidal and monothetic minimally almost periodic group, Fund. Math. 73 (1971/72), no. 2, 167–169.
S. Rolewicz, Some remarks on monothetic groups, Colloq. Math. XIII (1964), 28–29.
H. H. Schaefer, Topological Vector Spaces, The Macmillan Co., New York, Collier-Macmillan Ltd., London, 1966.
L. J. Sulley, On countable inductive limits of locally compact abelian groups, J. Lond.Math. Soc. (2) 5 (1972), 629–637.
M. Tkachenko, Self-duality in the class of precompact groups, Topology Appl. 156 (2009), no. 12, 2158–2165.
N. T. Varopoulos, Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 (1964), 465–516.
N.Y. Vilenkin, The theory of characters of topological Abelian groups with boundedness given (in Russian), Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 439–162.
A. Weil, L’intégration dans les groupes topologiques et ses applications, Actualités scientifiques et industrielles 869, Hermann & Cie, Paris, 1940.