Duality of locally quasi-convex convergence groups





continuous duality, convergence groups, local quasi-convexity, Pontryagin duality


In the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the Pontryagin duality is the continuous duality. We prove that local quasi-convexity is a necessary condition for a convergence group to be c-reflexive. Further, we prove that every character group of a convergence group is locally quasi-convex.


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Author Biography

Pranav Sharma, IIMT University

School of Basic Sciences


L. Außenhofer, Contributions to the Duality Theory of Abelian Topological Groups and to the Theory of Nuclear Groups, Dissertationes mathematicae. Institute of Mathematics, Polish Academy of Sciences, 1999.

W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Matheatics, Springer Berlin Heidelberg, 1991. https://doi.org/10.1007/BFb0089147

R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis, Bücher, Springer Netherlands, 2013.

M. Bruguera, Topological groups and convergence groups: Study of the Pontryagin duality, Thesis, 1999.

H.-P. Butzmann, Über diec-Reflexivität von Cc (X), Comment. Math. Helv. 47, no. 1 (1972), 92-101. https://doi.org/10.1007/BF02566791

H.-P. Butzmann, Duality theory for convergence groups, Topology Appl. 111, no. 1 (2000), 95-104. https://doi.org/10.1016/S0166-8641(99)00188-1

M. J. Chasco and E. Martín-Peinador, Binz-Butzmann duality versus Pontryagin duality, Arch. Math. (Basel) 63, no. 3 (1994), 264-270. https://doi.org/10.1007/BF01189829

M. J. Chasco, D. Dikranjan and E. Martín-Peinador, A survey on reflexivity of abelian topological groups, Topology Appl. 159, no. 9 (2012), 2290-2309. https://doi.org/10.1016/j.topol.2012.04.012

S. Dolecki and F. Mynard, Convergence Foundations of Topology, World Scientific Publishing Company, 2016. https://doi.org/10.1142/9012

E. Martín-Peinador, A reflexive admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123, no. 11 (1995), 3563-3566. https://doi.org/10.2307/2161108

E. Martín-Peinador and V. Tarieladze, A property of Dunford-Pettis type in topological groups, Proc. Amer. Math. Soc. 132, no. 6 (2004), 1827-1837. https://doi.org/10.1090/S0002-9939-03-07249-6

P. Sharma, Locally quasi-convex convergence groups, Topology Appl. 285 (2020), 107384. https://doi.org/10.1016/j.topol.2020.107384

P. Sharma and S. Mishra, Duality in topological and convergence groups, Top. Proc., to appear.




How to Cite

P. Sharma, “Duality of locally quasi-convex convergence groups”, Appl. Gen. Topol., vol. 22, no. 1, pp. 193–198, Apr. 2021.