Open subgroups, compact subgroups amd Binz-Butzmannn reflexivity
A number of attempts to extend Pontryagin duality theory to categories of groups larger than that of locally compact abelian groups have been made using different approaches. The extension to the category of topological abelian groups created the concept of reflexive group. In this paper we deal wit...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 1996 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/57058 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/57058 |
| Access Level: | acceso abierto |
| Palabra clave: | 515.1 Reflexive group Continuous convergence structure Character Dual group Topología 1210 Topología |
| Sumario: | A number of attempts to extend Pontryagin duality theory to categories of groups larger than that of locally compact abelian groups have been made using different approaches. The extension to the category of topological abelian groups created the concept of reflexive group. In this paper we deal with the extension of Pontryagin duality to the category of convergence abelian groups. Reflexivity in this category was defined and studied by E. Binz and H. Butzmann. A convergence group is reflexive (subsequently called BB-reflexive by us in our work) if the canonical embedding into the bidual is a convergence isomorphism. Topological abelian groups are, in an obvious way, convergence groups; therefore it is natural to compare reflexivity and BB-reflexivity for them. Chasco and Martín-Peinador (1994) show that these two notions are independent. However some properties of reflexive groups also hold for BB-reflexive groups, and the purpose of this paper is to show two of them. Namely, we prove that if an abelian topological group G contains an open subgroup A, then G is BB-reflexive if and only if A is BB-reflexive. Next, if G has sufficiently many continuous characters and K is a compact subgroup of G, then G is BB-reflexive if and only if G/K is BB-reflexive. |
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