Examples of locally compact noncompact minimal topological groups

Dierolf, Susanne; Schwanengel, Ulrich

doi:10.2140/pjm.1979.82.349

Cited by 55 publications

(32 citation statements)

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“…However, there is no reason why a 3-lemma diagram must also be a topological 3-lemma diagram in GT. However, that is exactly the meaning of the result of Dierolf and Schwanengel [9].…”

Section: Proposition 23 (Dierolf and Schwanengelmentioning

confidence: 85%

“…And since the right square is commutative and the arrows X 3 Z and X 1 3 Z are quotient maps, t X aY % taY. Thus, by the result of Dierolf and Schwanengel [9], t t X , which makes T open. h…”

Section: Proposition 31 (A 3-lemma For Topological Groups) Assume Tmentioning

confidence: 92%

“…This result can be seen as a particular case of a nice result of Dierolf and Schwanengel [9]. Recall that given a group topology t on a commutative group G and given a subgroup H then tjH means the topology induced on H by t while taH means the quotient topology on GaH (or the topology induced on the left or right cosets, in the case that H is not a normal subgroup of G).…”

mentioning

confidence: 86%

“…Recall that given a group topology t on a commutative group G and given a subgroup H then tjH means the topology induced on H by t while taH means the quotient topology on GaH (or the topology induced on the left or right cosets, in the case that H is not a normal subgroup of G). [9]). Let G be a group and H & G be a subgroup.…”

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confidence: 99%

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On the "three-space" problem for locally quasi-convex topological groups

Castillo

2000

Archiv der Mathematik

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We study, in the context of abelian topological groups, the ªthree-spaceº problem for the property of being locally quasi-convex, after a paper of M. Bruguera. Our main contributions are: establishing a 3-lemma suitable to work with topological groups (which allows to translate the basic elements of homological algebra to the category of topological groups) and obtaining the analogue, for topological groups, of Dierolfs result in topological vector spaces:Theorem. Given two abelian locally quasi-convex groups H and G there exists a nonlocally-quasi-convex extension of H and G if and only if there exists a non-locally-quasiconvex extension of S (the circle group) and G.

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“…However, there is no reason why a 3-lemma diagram must also be a topological 3-lemma diagram in GT. However, that is exactly the meaning of the result of Dierolf and Schwanengel [9].…”

Section: Proposition 23 (Dierolf and Schwanengelmentioning

confidence: 85%

Section: Proposition 31 (A 3-lemma For Topological Groups) Assume Tmentioning

confidence: 92%

mentioning

confidence: 86%

mentioning

confidence: 99%

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On the "three-space" problem for locally quasi-convex topological groups

Castillo

2000

Archiv der Mathematik

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“…We cannot help mentioning that satisfying the equation L(X, ℓ 2 ) = Π 1 (X, ℓ 2 ) is not only stable by products but also a 3-space property (see [20]). At this point it might be interesting to look at paper [19]: there, a very nice result of Dierolf and Schwanengel [29] is revisited to show that the 3-lemma (hence, the notion of equivalent sequences) still works well for topologically exact sequences of topological groups (even whithout the open mapping theorem). This allows one to transport the basic elements of homological algebra (pullback and push-out constructions) to abelian topological groups.…”

Section: Lemma 719 ([65]) a Banach Space X Is A Kalton-pełczyński Smentioning

confidence: 99%

Banach space techniques underpinning a theory for nearly additive mappings

Sánchez

Castillo

2002

Dissertationes Math.

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Ulam's stability problem for additive maps and the theory of extensions of Banach spaces have remained so far unaware of each other. To support this assertion it is enough to peruse the survey articles [36,52] In fact, both worlds constitute two faces of the same coin. In this spirit, the purpose of this paper is to progress towards a global understanding of the problem, and we shall do so by developing a theory of vector-valued nearly additive mappings on semigroups. While a unified theory seems to be out of reach by now, the part of the theory involving quasi-normed groups and maps with values in Banach spaces seems to be ripe with these pages. We hasten to alert the reader that this paper can by no means be considered a survey, although it contains occasional historical comments; two excellent survey papers are [36,52].

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M‐normal subgroups and non‐abelian lower continuous topological groups

Peng

2021

Mathematische Nachrichten

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In this paper, we introduce the notion of m-normal subgroups and show that the m-normal subgroups of connect locally compact groups and that of compact groups are closely related to Lie groups. As applications, we extend Theorem 3.8 in [D. Peng and W. He, Lower continuous topological groups, Topol. Appl. 265 (2019)] by giving a lower continuity criterion for dense subgroups of arbitrary topological groups. It is shown that lower continuity is preserved under taking topological products. It is also shown that an infinite compact group is hereditarily lower continuous if and only if the normalizer of every non-trivial finite subgroup is finite.

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Examples of locally compact noncompact minimal topological groups

Cited by 55 publications

References 1 publication

On the "three-space" problem for locally quasi-convex topological groups

On the "three-space" problem for locally quasi-convex topological groups

Banach space techniques underpinning a theory for nearly additive mappings

M‐normal subgroups and non‐abelian lower continuous topological groups

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