The Bohr compactification, modulo a metrizable subgroup

Comfort, W. W.; Trigos-Arrieta, F. Javier; Wu, Shun

doi:10.4064/fm_1993_143_2_1_119_136

Cited by 18 publications

(23 citation statements)

References 14 publications

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“…We refer the reader for a detailed proof in the case A = Z to [29], [30]. When A = k<ω Z(p k ) or A = Z(p ∞ ) there are, according to Theorem 5.1 or 5.2 respectively, a sequence (x n ) in A and a faithfully indexed family {h ξ : ξ < c} ⊆ Hom(A, T) such that h ξ (x n ) → 0 for each ξ < c. It is then clear, as in [9], that with H := {h ξ : ξ < c} ⊆ Hom(A, T) we have h(x n ) → 0 for each h ∈ H.…”

Section: Topologies With Convergent Sequencesmentioning

confidence: 99%

“…Theorem 1.6. [9]. Let G be an infinite Abelian group with K := Hom(G, T) = G d , let (x n ) n be a faithfully indexed sequence in G, and let…”

Section: Introductionmentioning

confidence: 99%

“…[22](24.25)), then a reference to Theorem 1.4 completes the proof: the condition λ(A) > 0 would imply A = Hom(G, T), so that x n → 0 in the Bohr topology of G d , contrary to the theorem of Leptin and Glicksberg cited above. We refer the reader to [9] for the proof (for general Abelian G) that λ(A) = 0.…”

Section: Introductionmentioning

confidence: 99%

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Making group topologies with, and without, convergent sequences

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Section: Topologies With Convergent Sequencesmentioning

confidence: 99%

“…Theorem 1.6. [9]. Let G be an infinite Abelian group with K := Hom(G, T) = G d , let (x n ) n be a faithfully indexed sequence in G, and let…”

Section: Introductionmentioning

confidence: 99%

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2006

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“…Remark 1.8. According to [CTW,Lemma 3.10], if X is an infinite compact abelian group and v ∈ X N contains a faithfully indexed subsequence, then s v (X) has zero Haar measure, so the index [X : s v (X)] is uncountable, as X has measure 1.…”

Section: About Borel Complexity Of Characterized Subgroupsmentioning

confidence: 99%

On the Borel complexity of characterized subgroups

Dikranjan¹,

Impieri²

2016

Topology and its Applications

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In a compact abelian group X, a subgroup H is called characterized if there exists a sequence of characters v = (v n ) of X such that H = {x ∈ X : v n (x) → 0 in T}. Gabriyelyan proved for X = T, that the characterized subgroup {x ∈ T : n!x → 0 in T} is not an F σ -set. In this paper, we obtain a complete description of the F σ -subgroups of T characterized by sequences of integers v = (v n ) such that v n |v n+1 for all n ∈ N showing that these are exactly the countable subgroups of T. Moreover, in the general setting of compact metrizable abelian groups, we give a new point of view to study the Borel complexity of characterized subgroups in terms of appropriate test-topologies defined on the group X.

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“…To finish the paper we deal with the property of strongly respecting compactness introduced in [13]. Comfort, Trigos-Arrieta and Wu proved that every abelian locally compact group strongly respects compactness.…”

Section: Introductionmentioning

confidence: 99%

A dichotomy property for locally compact groups

Ferrer

Hernández

Tárrega

2018

Journal of Functional Analysis

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We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of ℓ 1 . For that purpose, we transfer to general locally compact groups the notion of interpolation (I 0 ) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence {g n } n<ω in a locally compact group G, then either {g n } n<ω has a weak Cauchy subsequence or contains a subsequence that is an I 0 set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group G, an old question that remains open since 1974 (see [32] and [20]). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this property for abelian locally compact groups.

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The Bohr compactification, modulo a metrizable subgroup

Cited by 18 publications

References 14 publications

Making group topologies with, and without, convergent sequences

Making group topologies with, and without, convergent sequences

On the Borel complexity of characterized subgroups

A dichotomy property for locally compact groups

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