Abelian groups which satisfy Pontryagin duality need not respect compactness

Remus, Dieter; Trigos-Arrieta, F. Javier

doi:10.1090/S0002-9939-1993-1132422-4

Abelian groups which satisfy Pontryagin duality need not respect compactness
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by Dieter Remus and F. Javier Trigos-Arrieta

Proc. Amer. Math. Soc. 117 (1993), 1195-1200

DOI: https://doi.org/10.1090/S0002-9939-1993-1132422-4

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Abstract:

Let ${\mathbf {G}}$ be a topological Abelian group with character group ${{\mathbf {G}}^ \wedge }$. We will say that ${\mathbf {G}}$ respects compactness if its original topology and the weakest topology that makes each element of ${{\mathbf {G}}^ \wedge }$ continuous produce the same compact subspaces. We show the existence of groups which satisfy Pontryagin duality and do not respect compactness, thus furnishing counterexamples to a result published by Venkataraman in 1975. Our counterexamples will be the additive groups of all reflexive infinite-dimensional real Banach spaces. In order to do so, we first characterize those locally convex reflexive real spaces whose additive groups respect compactness. They are exactly the Montel spaces. Finally, we study the class of those groups that satisfy Pontryagin duality and respect compactness.

References

Ichiro Amemiya and Yukio K\B{o}mura, Über nicht-vollständige Montelräume, Math. Ann. 177 (1968), 273–277 (German). MR 232182, DOI 10.1007/BF01350719
N. Bourbaki, Topological vector spaces. Chapters 1–5, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1987. Translated from the French by H. G. Eggleston and S. Madan. MR 910295, DOI 10.1007/978-3-642-61715-7
Irving Glicksberg, Uniform boundedness for groups, Canadian J. Math. 14 (1962), 269–276. MR 155923, DOI 10.4153/CJM-1962-017-3
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
Edwin Hewitt and H. S. Zuckerman, A group-theoretic method in approximation theory, Ann. of Math. (2) 52 (1950), 557–567. MR 41142, DOI 10.2307/1969432
Samuel Kaplan, Extensions of the Pontrjagin duality. I. Infinite products, Duke Math. J. 15 (1948), 649–658. MR 26999
Samuel Kaplan, Extensions of the Pontrjagin duality. II. Direct and inverse sequences, Duke Math. J. 17 (1950), 419–435. MR 49906
Yukio K\B{o}mura, Some examples on linear topological spaces, Math. Ann. 153 (1964), 150–162. MR 185417, DOI 10.1007/BF01361183
Horst Leptin, Zur Dualitätstheorie projektiver Limites abelscher Gruppen, Abh. Math. Sem. Univ. Hamburg 19 (1955), 264–268 (German). MR 68544, DOI 10.1007/BF02988876
N. Noble, $k$-groups and duality, Trans. Amer. Math. Soc. 151 (1970), 551–561. MR 270070, DOI 10.1090/S0002-9947-1970-0270070-8
Marianne Freundlich Smith, The Pontrjagin duality theorem in linear spaces, Ann. of Math. (2) 56 (1952), 248–253. MR 49479, DOI 10.2307/1969798
F. Javier Trigos-Arrieta, Pseudocompactness on groups, General topology and applications (Staten Island, NY, 1989) Lecture Notes in Pure and Appl. Math., vol. 134, Dekker, New York, 1991, pp. 369–378. MR 1142814
F. Javier Trigos-Arrieta, Continuity, boundedness, connectedness and the Lindelöf property for topological groups, Proceedings of the Conference on Locales and Topological Groups (Curaçao, 1989), 1991, pp. 199–210. MR 1100517, DOI 10.1016/0022-4049(91)90018-W
Rangachari Venkataraman, Compactness in abelian topological groups, Pacific J. Math. 57 (1975), no. 2, 591–595. MR 387491
Rangachari Venkataraman, Extensions of Pontryagin duality, Math. Z. 143 (1975), no. 2, 105–112. MR 374335, DOI 10.1007/BF01187051
Rangachari Venkataraman, Interval of group topologies satisfying Pontryagin duality, Math. Z. 155 (1977), no. 2, 143–149. MR 447463, DOI 10.1007/BF01214214