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“Varopoulos paradigm”: Mackey property versus metrizability in topological groups


The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: “A metrizable locally convex space carries its Mackey topology ”. This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (Außenhofer and de la Barrera Mayoral in J Pure Appl Algebra 216(6):1340–1347, 2012; Díaz Nieto and Martín Peinador in Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics and Statistics, Vol 80 doi:10.1007/978-3-319-05224-3_7, 2014; Dikranjan et al. in Forum Math 26:723–757, 2014). We say that an abelian group G satisfies the Varopoulos paradigm (VP) if any metrizable locally quasi-convex topology on G is the Mackey topology. In the present paper we prove that in any unbounded group there exists a lqc metrizable topology that is not Mackey. This statement (Theorem C) allows us to show that the class of groups satisfying VP coincides with the class of finite exponent groups. Thus, a property of topological nature characterizes an algebraic feature of abelian groups.

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The authors sincerely thank the referee for his careful reading of our paper. His observations have contributed to improve and clarify some points in our previous version.

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Correspondence to E. Martín-Peinador.

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This paper is dedicated to George Mackey on the centenary of his birth (1916-2006)

The third named author was supported by the grant “Progetti di Eccellenza 2011/12” of Fondazione CARIPARO.

The second and the fourth named authors were partially supported by Ministerio de Economía y Competitividad grant: MTM2013-42486-P and MTM2016-79422-P.

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Außenhofer, L., de la Barrera Mayoral, D., Dikranjan, D. et al. “Varopoulos paradigm”: Mackey property versus metrizability in topological groups. Rev Mat Complut 30, 167–184 (2017).

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