Abstract
We define GP-nuclear groups as topological Abelian groups for which the groups of summable and absolutely summable sequences are the same algebraically and topologically. It is shown that in the metrizable case only the algebraic coincidence of the mentioned groups is needed for GP-nuclearity. Some permanence properties of the class of GP-nuclear groups are obtained. Our final result asserts that nuclear groups in the sense of Banaszczyk are GP-nuclear. The validity of the converse assertion remains open.
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References
L. Ausenhofer, Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups. Ph.D. Thesis, Tübingen University (Tübingen, 1998) (to appear in Dissertationes Math.)
W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math., 1466 Springer-Verlag (Berlin-Heidelberg, 1991).
W. Banaszczyk, Closed subgroups of nuclear spaces are weakly closed, Studia Math., 80 (1984), 119–128.
W. Banaszczyk, Polar lattices from the point of view of nuclear spaces, Rev. Mat. Univ. Complut. Madrid 2 (special issue) (1989), 35–46.
W. Banaszczyk, Pontryagin duality for subgroups and quotients of nuclear spaces, Math. Ann., 273 (1986), 653–664.
W. Banaszczyk, Summable families in nuclear groups, Studia Math., 105 (1993), 271–282.
W. Banaszczyk and V. Tarieladze, The strong Lévy property and related questions for nuclear groups, Preprint, 1999.
W. Banaszczyk and E. Martín-Peinador, Weakly pseudocompact subsets of nuclear groups, J. Pure Appl. Algebra, 138 (1999), 99–106.
N. Bourbaki, Éléments de mathématique: Topologie générale, Diffusion C. C. L. S. (Paris, 1971).
M. J. Chasco, E. Martín-Peinador and V. Tarieladze, On Mackey topology for groups, Studia Mathematica, 132 (1999), 257–284.
C. Constantinescu, Spaces of Measures, De Gruyter studies in Mathematics, 4. De Gruyter (Berlin-New York, 1984).
X. Domíngnez and V. Tarieladze, The Schur property for topological groups, Preprint, 1998.
A. Dvoretzky, On series in linear topological spaces, Israel J. Math., 1 (1963), 37–47.
J. Galindo, Structure and analysis on nuclear groups, Preprint, 1997.
H. Jarchow, Locally Convex Spaces, B. G. Teubner (Stuttgart, 1981).
S. Kaplan, Extensions of the Pontrjagin duality I: infinite products, Duke Math. J., 15 (1948), 649–658.
A. Pietsch, Nuclear Locally Convex Spaces, Springer-Verlag (1972).
S. Rolewìcz, Metric Linear Spaces, Reidel Publishing Company and Polish Scientific Publishers (1985).
H. H. Schaefer, Topological Vector Spaces, Springer-Verlag (1970).
V. Tarieladze, On Ito-Nisio type theorems for DS-groups, Georgian Mathematical Journal, 4 (1977), 477–500.
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Domínguez, X., Tarieladze, V. Nuclear and GP-Nuclear Groups. Acta Mathematica Hungarica 88, 301–322 (2000). https://doi.org/10.1023/A:1026728123531
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DOI: https://doi.org/10.1023/A:1026728123531