View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by EPrints Complutense UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS MATEMÁTICAS TESIS DOCTORAL Duality on abelian topological groups: the Mackey problema Dualidad en grupos abelianos topológicos : el problema Mackey MEMORIA PARA OPTAR AL GRADO DE DOCTOR PRESENTADA POR Daniel de la Barrera Mayoral Directoras Lydia Außenhofer Elena Martín-Peinador Madrid, 2016 © Daniel de la Barrera Mayoral, 2015 UNIVERSIDAD COMPLUTENSE DE MADRID Facultad de Ciencias Matematicas´ Duality on abelian topological groups: The Mackey Problem. (Dualidad en grupos abelianos topologicos:´ El Problema Mackey.) Daniel de la Barrera Mayoral. Memoria presentada para optar al grado de Doctor con mencion´ de Doctor Europeus en Matematicas,´ elaborada bajo la direccion´ de: Prof. Lydia Außenhofer Prof. Elena Mart´ın-Peinador Agradecimientos. En primer lugar, por supuesto, a mi familia, para los cuales´ es un honor que haya llegado a doctorarme. Tambien,´ tengo que agradecer a Elena y a Lydia, por estos seis anos˜ de mi vida que nunca podre´ olvidar. Por todas las conferencias en las que Elena ha estado dando su apoyo moral entre el publico.´ A Lydia, por esa semana que pasamos en Eichstatt¨ y dieron lugar a nuestro primer art´ıculo conjunto. Tambien´ debo agradecer a los miembros del departamento por ese companerismo˜ del que hacen gala y por la posibilidad de iniciarme en ese arte que se llama do- cencia universitaria. Por supuesto, debo agradecer al Ministerio de Educacion´ por las becas conce- didas en mis anos˜ de Licenciatura y posteriormente por los Proyectos MTM2013- 42486-P and MTM2009-14409-C02-01, que me han permitido participar en var- ios congresos y viajar a Udine. A Vaja le agradezco las largas y gratas conversaciones matematicas´ en sus visitas a la Universidad Complutense. A Ma Jesus,´ Xabier y Salvador les debo tambien´ un gran apoyo en mis diversos viajes por la geograf´ıa espanola.˜ A Dikran, Anna, Guisepina y Hans, por haberme acogido en el Departamento de Informatica´ y Matematica´ de la Universidad de Udine durante mi estancia en Italia. Especialmente a Dikran, por los viajes en tierras italianas que hicimos jun- tos y las matematicas´ que aprend´ı all´ı. Tampoco debo olvidar a Marco, Joaqu´ın, Mar´ıa y Merche, quienes fueron los primeros en conocerme dentro de la facultad con las Olimpiadas Matematicas´ y Concursos de Primavera. Al P. Emilio, debo reconocerle el merito´ de ser el primero en darse cuenta que podr´ıa llegar lejos en matematicas´ cuando aun´ andaba por Secundaria. A Santi, Miguel y David por compartir aquellos´ maravillosos anos˜ de licen- ciatura. A Santi, Jesus´ y Sergio, les agradezco que, sin ser doctores, se hayan II le´ıdo mi tesis. A mis amigos, tengo que agradecerles que me aguanten completa- mente y sin dudarlo: Mar´ıa, Miguel, Vero, Alvaro,´ Estrella, Antonio, Alberto, va por vosotros. Paco y Angel,´ esos hombres de negro que siempre estan´ dispuestos a echar una mano. Contents Introduction V Introducción XIX 1 Definitions, notation and basic results. 1 1.1 Topological groups. 1 1.2 Precompact topologies. 11 1.3 Topologies of uniform convergence. 12 1.4 T-sequences and complete topologies. 14 2 D-sequences. 17 2.1 Main definitions. 17 2.2 Numerical results concerning D-sequences. 20 2.3 The groups Z(b1). 28 2.4 The group Zb of b-adic integers. 29 3 The Mackey problem. 33 3.1 Historical background. 33 3.2 Compatible group topologies. 36 3.3 Mackey topology and open subgroups. 38 3.4 Alternative statements for the Mackey Problem. 40 3.5 G-Mackey problem. 40 4 The group of integers 43 4.1 Different topologies on the integers. 43 4.2 Precompact topologies, p-adic topologies and b-adic topologies. 44 IV CONTENTS 4.3 Topologies of uniform convergence on Z. 52 4.3.1 Topologies of uniform convergence on Z induced by a D- sequence. 53 ` 4.3.2 Duality if b 2 D1. ..................... 60 ` 4.3.3 Duality if b < D1. ..................... 70 4.3.4 Supremum of topologies of uniform convergence on D- sequences. 74 4.4 The topology δb. .......................... 76 4.4.1 δb if b has bounded ratios. 79 ` 4.4.2 δb if b 2 D1 for some `................... 83 4.5 Complete topologies on Z. ..................... 85 4.6 Open questions and problems. 86 5 The unit circle and its subgroups. 89 5.1 Two different topologies on Z(b1) . 89 5.1.1 The usual topology on Z(b1). 89 5.1.2 Topologies of uniform convergence on Z(b1). 91 5.1.3 Comparing ηb and τU . ................... 96 1 5.2 Computing the dual group of (Z(b ); ηb). 97 6 The Varopoulos Paradigm. 103 6.1 Introduction. 103 6.2 Proofs on bounded groups: (i) ) (ii) in Theorem 6.1.3. 104 6.3 Proofs on unbounded groups: (ii) ) (i) in Theorem 6.1.3. 111 6.4 Some corollaries of Theorem 6.2.1. 117 Index 121 Bibliography 124 Introduction. The idea of a general theory of continuous groups is due to S. Lie, who devel- oped his theory in the decade 1874-1884. Lie’s work is the origin of both modern theory of Lie groups and the general theory of topological groups. However, the topological considerations that are nowadays essential in both theories, are not part of his work. A topological point of view in the theory of continuous groups was first in- troduced by Hilbert. Precisely, in his famous list of 23 problems, presented in the International Congress of Mathematicians of 1900, held in Paris, the Fifth Problem boosted investigations on topological groups. In 1932, Stefan Banach, defined in Theorie` des Operations` Lineaires,` the spaces that would be named after him as special cases of topological groups. Since then, both theories (Banach spaces and topological groups) had been developed in a different, but somehow parallel, way. Several theorems for Banach, or even for locally convex spaces have been reformulated for abelian topological groups [BCMPT01], [BMPT04], [CHT96], [HGM99], [MPT04]. The main obstacle for this task is the lack of the notion of convexity for topological groups due to the lack of an outer operation. However, Vilenkin gave the definition of quasi-convex subset for a topological abelian group inspired in the Hahn-Banach Theorem. In order to deal with it, many convenient tools had to be developed and this opened the possibility of a fruitful treatment of topological groups. The notion of quasi-convexity depends on the topology, in contrast with convexity, which is a purely algebraic notion. With quasi-convex subsets at hand, it was quite natural to define locally quasi- convex groups, which was also done by Vilenkin in [Vil51]. In the sequel, we will only deal with abelian groups, even if we do not explic- itly mention it. VI INTRODUCTION. Duality theory for locally convex spaces was mainly developed in the mid of twentieth century, and by now it is a well-known rich theory. There is a very natural way to extend it to locally quasi-convex abelian groups. Fix first the dualizing object as the unit circle of the complex plane T. The con- tinuous homomorphisms from a topological group into T play the role of the con- tinuous linear forms and they are called continuous characters. The set of continuous characters defined on a group G has a natural group structure provided by the group structure of T. Thus, we can speak of the group of continuous characters of G, which is called the dual group of G and it is the analogous notion to the dual space in the topological vector space setting. The natural topology in the dual group is the compact open topology, because of the Pontryagin Duality Theorem which holds for locally compact groups, the most distinguished class of topological groups. The research we have done in this memory deals with the following problem, which we have denominated the Mackey Problem: The Mackey Problem: Let (G; τ) be an abelian topological group and let G^ be its dual group. Consider the set of all locally quasi-convex group topologies in G whose dual group coincides with G^. They will be called compatible topolo- gies. It is known that there is a minimum for this set, which is the weak topology induced by G^. Does there exist a maximum element in this set? Current state of the problem: For an abelian topological group G, it is known that there is always at least one locally quasi-convex compatible topology, namely the weak topology, which is always precompact. In general, it is not known if the set of compatible locally quasi-convex group topologies has a maximum element; whenever it exists, it will be called the Mackey topology for (G; τ). The problem of finding the largest compatible locally quasi-convex topology for a given topological group G, with this degree of generality (i.e. in the framework of locally quasi-convex groups) was first stated in 1999 in [CMPT99]. Previously in 1964 Varopoulos [Var64] had studied the question for the class of locally precompact group topologies. Partial answers are the following: if G is locally quasi-convex complete and metrizable then there exists the Mackey topology [CMPT99] and it coincides with INTRODUCTION. VII the original one. If G is locally quasi-convex metrizable (but not complete) the original topology may not coincide with the Mackey topology [DMPT14]. If G is a locally compact abelian group, its original topology is locally quasi- convex and it coincides with the Mackey topology. It is proved in [ADMP15] that the set of all locally quasi-convex compatible group topologies has cardinal greater or equal than 3.