Dualities of locally compact modules over the rationals

Dikranjan, Dikran; Milan, Chiara

doi:10.1016/s0021-8693(02)00115-1

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…Let T and ϕ be as in Lemma 4.5. Since T is a sequence converging to 0 in R, and S is a sequence converging to 0 in K, it follows that (9). From θ({0} × S) ⊆ X, we conclude that X ∩ ker f is qc-dense in ker f .…”

Section: Proof Of Theorem 27mentioning

confidence: 82%

Quasi-convex density and determining subgroups of compact abelian groups

Dikranjan¹,

Shakhmatov²

2010

Journal of Mathematical Analysis and Applications

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For an abelian topological group G let G^* denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G) and an open neighbourhood U of 0 in the circle group, we show that the set of all characters which send X into U has the same size as G^*. (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the restriction homomorphism G^* --> D^* is an isomorphism between G^* and D^*. We prove that w(G) = min {|D|: D is a subgroup of G that determines G} for every infinite compact abelian group G. In particular, an infinite compact abelian group determined by a countable subgroup is metrizable. This gives a negative answer to questions of Comfort, Hernandez, Macario, Raczkowski and Trigos-Arrieta. As an application, we furnish a short elementary proof of the result from [13] that a compact abelian group G is metrizable provided that every dense subgroup of G determines G.Comment: 11 pages. The proof of Lemma 3.6 (from version 2) has been significantly simplified and shorten. 2 lemmas and 4 references have been added. The order of the material has been substantially changed as wel

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Section: Proof Of Theorem 27mentioning

confidence: 82%

Quasi-convex density and determining subgroups of compact abelian groups

Dikranjan¹,

Shakhmatov²

2010

Journal of Mathematical Analysis and Applications

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“…By Exercise 7.17 Q/Z ∼ = p J p is totally disconnected, so by Corollary 6.21 H has no surjective characters χ : H → T. Now let χ ∈ K be non-zero. The compact group Q is closely related to the adele rings of the field Q, more detail can be found in [34,38,75,97].…”

Section: Some General Properties Of the Dualmentioning

confidence: 99%

“…The first example of a discontinuous duality was given in [29,Theorem 11.1]. Discontinuous dualities of L Q and its subcategories are discussed in [34]. It was conjectured by Prodanov that in case R is an algebraic number ring uniqueness of dualities is available if and only if R is a principal ideal domain.…”

Section: Uniqueness Of Pontryagin-van Kampen Dualitymentioning

confidence: 99%

Introduction to topological groups

Diestel¹,

Spalsbury²

2014

Graduate Studies in Mathematics

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“…Let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$K={\mathbb R}\times H$\end{document} and u = (1, v ) ∈ K . Then the cyclic subgroup 〈 u 〉 of K is discrete and the quotient group C = K /〈 u 〉 is isomorphic to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{{\mathbb Q}}$\end{document}; see [8, Section 2.1]. Therefore, it suffices to prove that C a contains a sequence converging to 0 C that is qc‐dense in C .…”

Section: A Qc‐dense Super‐sequence In the Arc Component Of \Documentcmentioning

confidence: 99%

Quasi‐convexly dense and suitable sets in the arc component of a compact group

Dikranjan¹,

Shakhmatov²

2011

Mathematische Nachrichten

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We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super-sequence converging to 0 that is qc-dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism r : G → G_a is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component Ga contains a super-sequence converging to the identity that is qc-dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group

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Dualities of locally compact modules over the rationals

Cited by 4 publications

References 10 publications

Quasi-convex density and determining subgroups of compact abelian groups

Quasi-convex density and determining subgroups of compact abelian groups

Introduction to topological groups

Quasi‐convexly dense and suitable sets in the arc component of a compact group

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