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Martin Peinador, E. and Taieladze, V. (2016) Mackey topology on locally convex spaces and on locally quasiconvex groups. Similarities and historical remarks. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales  Serie A: Matematicas, 110 (2). pp. 667679. ISSN 15787303

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Official URL: http://link.springer.com/article/10.1007/s1339801502560
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Abstract
A counterpart of the Mackey–Arens Theorem for the class of locally quasiconvex topological Abelian groups (LQCgroups) was initiated in Chasco et al. (Stud Math 132(3):257–284, 1999). Several authors have been interested in the problems posed there and have done clarifying contributions, although the main question of that source remains open. Some differences between the Mackey Theory for locally convex spaces and for locally quasiconvex groups, stem from the following fact: The supremum of all compatible locally quasiconvex topologies for a topological abelian group G may not coincide with the topology of uniform convergence on the weak quasiconvex compact subsets of the dual groupG∧. Thus, a substantial part of the classical Mackey–Arens Theorem cannot be generalized to LQCgroups. Furthermore, the mentioned fact gives rise to a grading in the property of “being a Mackey group”, as defined and thoroughly studied in Díaz Nieto and MartínPeinador (Proceedings in Mathematics and Statistics 80:119–144, 2014). At present it is not known—and this is the main open question—if the supremum of all the compatible locally quasiconvex topologies on a topological group is in fact a compatible topology. In the present paper we do a sort of historical review on the Mackey Theory, and we compare it in the two settings of locally convex spaces and of locally quasiconvex groups. We point out some general questions which are still open, under the name of Problems.
Item Type:  Article 

Uncontrolled Keywords:  Compatible topology; Locally convex space; Locally quasiconvex group; Mackey topology; Prelocally compact group; Precompact group. 
Subjects:  Sciences > Mathematics > Functional analysis and Operator theory Sciences > Mathematics > Algebraic geometry Sciences > Mathematics > Topology 
ID Code:  39367 
Deposited On:  10 Oct 2016 12:24 
Last Modified:  12 Dec 2018 15:12 
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