Strong Reflexivity of Abelian Groups
Montserrat Bruguera2001
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Abstract
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This paper explores the concept of strong reflexivity within the framework of Abelian groups, particularly focusing on topological and convergence groups. It defines the dual and bidual groups of Abelian topological groups, introducing the notion of BB-reflexivity and its implications for closed subgroups and Hausdorff quotients. The results demonstrate that BB-strongly reflexive groups possess desirable properties regarding the stability under group operations and relationships between duals.
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An Abelian topological group is called strongly reflexive if every closed subgroup and every Hausdorff quotient of the group and of its dual group are reflexive. In the class of locally compact Abelian groups (LCA) there is no need to define "strong reflexivity": it does not add anything new to reflexivity, which by the Pontryagin - van Kampen Theorem is known to hold for every member of the class. In this survey we collect how much of "reflexivity" holds for different classes of groups, with especial emphasis in the classes of pseudocompact groups, $\omega$-groups and $P$-groups, in which some reflexive groups have been recently detected. In section 3.5 we complete the duality relationship between the classes of $P$-groups and $\omega$-bounded groups.
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An Abelian topological group is called strongly reflexive if every closed subgroup and every Hausdorff quotient of the group and of its dual group are reflexive. In the class of locally compact Abelian groups (LCA) there is no need to define "strong reflexivity": it does not add anything new to reflexivity, which by the Pontryagin-van Kampen Theorem is known to hold for every member of the class. In this survey we collect how much of "reflexivity" holds for different classes of groups, with especial emphasis in the classes of pseudocompact groups, ω-groups and P-groups, in which some reflexive groups have been recently detected. In section 3.5 we complete the duality relationship between the classes of P-groups and ω-bounded groups, already outlined in [26]. By no means we can claim completeness of the survey: just an ordered view of the topic, with some small new results indicated in the text.
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It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book "Abstract Harmonic Analysis" [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups. Walter Rudin's book "Fourier Analysis on Groups" [2] includes an elegant proof of the Pontryagin-van Kampen Duality Theorem. Much gentler than these is "Introduction to Topological Groups" [3] by Taqdir Husain which has an introduction to topological group theory, Haar measure, the Peter-Weyl Theorem and Duality Theory. Of course the book "Topological Groups" [4] by Lev Semyonovich Pontryagin himself was a tour de force for its time. P. S. Aleksandrov, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko described this book in glowing terms: "This book belongs to that rare category of mathematical works that can truly be called classical-books which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians". The final book I mention from my graduate studies days is "Topological Transformation Groups" [5] by Deane Montgomery and Leo Zippin which contains a solution of Hilbert's fifth problem as well as a structure theory for locally compact non-abelian groups. These five books gave me a good feeling for the most significant research on locally compact group theory in the first 60 years of the twentieth century. My own contribution to understanding the structure of locally compact abelian groups was a small book "Pontryagin Duality and the Structure of Locally Compact Abelian Groups" [6] which was translated into Russian and served to introduce a generation of young Soviet mathematicians to this topic. Far from locally compact groups, A.A. Markov [7,8] introduced the study of free topological groups. This was followed up by M.I. Graev in 1948 [9] with a slightly more general concept. Free topological groups are an analogue of free groups in abstract group theory. Markov gave a very long construction of the free topological group on a Tychonoff space and also proved its uniqueness. Graev's proof is also long. Shorter proofs appeared after a few years. Today one derives the existence of Markov and Graev free topological groups from the Adjoint Functor Theorem. Free topological groups have been an active area of research to this day, especially by Alexander Vladimirovich Arhangel'skii of Moscow State University and his former doctoral students and they have produced a wealth of deep and interesting results. Now let me turn to this volume. My aim for "Topological Groups: Yesterday, Today, Tomorrow" is for these articles to describe significant topics in topological group theory in the 20th century and the early 21st century as well as providing some guidance to the future directions topological group theory might take by including some interesting open questions.
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