Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Pontryagin duality theorem; dual group; convergence group; continuous convergence; reflexive group; strong reflexive group; k-space; Čech complete group; k-group
Pontryagin duality theorem; dual group; convergence group; continuous convergence; reflexive group; strong reflexive group; k-space; Čech complete group; k-group
Summary:
A reflexive topological group $G$ is called strongly reflexive if each closed subgroup and each Hausdorff quotient of the group $G$ and of its dual group is reflexive. In this paper we establish an adequate concept of strong reflexivity for convergence groups. We prove that complete metrizable nuclear groups and products of countably many locally compact topological groups are BB-strongly reflexive.
References:
[1] W. Banaszczyk: Additive Subgroups of Topological Vector Spaces. Lecture Notes in Mathematics 1466. Springer-Verlag, Berlin-Heidelberg-New York, 1991. MR 1119302
[2] E. Binz: Continuous Convergence in $C(X)$. Lecture Notes in Mathematics 469. Springer-Verlag, Berlin-Heidelberg-New York, 1975. MR 0461418
[3] R. Brown, P. J. Higgins and S. A. Morris: Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties. Math. Proc. Camb. Phil. Soc. 78 (1975), 19–32. DOI 10.1017/S0305004100051483 | MR 0453915
[4] H. P. Butzmann: Pontryagin duality for convergence groups of unimodular continuous functions. Czechoslovak Math. J. 33 (1983), 212–220. MR 0699022 | Zbl 0528.54005
[5] H. P. Butzmann: $c$-Duality Theory for Convergence Groups. Lecture in the Course “Convergence and Topology”. Erice, 1998.
[6] M. J. Chasco: Pontryagin duality for metrizable groups. Arch. Math. 70 (1998), 22–28. DOI 10.1007/s000130050160 | MR 1487450 | Zbl 0899.22001
[7] M. J. Chasco and E. Martín-Peinador: Binz-Butzmann duality versus Pontryagin duality. Arch. Math. 63 (1994), 264–270. DOI 10.1007/BF01189829 | MR 1287256
[8] C. H. Cook and H. R. Fischer: On equicontinuity and continuous convergence. Math. Ann. 159 (1965), 94–104. DOI 10.1007/BF01360283 | MR 0179752
[9] H. R. Fischer: Limesräume. Math. Ann. 137 (1959), 269–303. MR 0109339 | Zbl 0086.08803
[10] S. Kaplan: Extensions of the Pontryagin duality I: Infinite products. B. G. Duke Math. 15 (1948), 649–658. MR 0026999
[11] H. Leptin: Bemerkung zu einem Satz von S. Kaplan. B. G. Arch. der Math. 6 (1955), 264–268. MR 0066397 | Zbl 0065.01601
[12] E. Martín-Peinador: A reflexive admissible topological group must be locally compact. Proc. Amer. Math. Soc. 123 (1995), 3563–3566. DOI 10.2307/2161108 | MR 1301516
[13] N. Noble: $K$-groups and duality. Trans. Amer. Math. Soc. 151 (1970), 551–561. MR 0270070 | Zbl 0229.22012
[14] N. Roelcke and S. Dierolf: Uniform Structures on Topological Groups and Their Quotients. Advanced Book Program. McGraw-Hill International Book Company, 1981. MR 0644485
Search
Partner of
