Quasi-homomorphisms

Félix Cabello Sánchez. "Quasi-homomorphisms." Fundamenta Mathematicae 178.3 (2003): 255-270. <http://eudml.org/doc/283225>.

@article{FélixCabelloSánchez2003,
abstract = { We study the stability of homomorphisms between topological (abelian) groups. Inspired by the "singular" case in the stability of Cauchy's equation and the technique of quasi-linear maps we introduce quasi-homomorphisms between topological groups, that is, maps ω: 𝒢 → ℋ such that ω(0) = 0 and ω(x+y) - ω(x) - ω(y) → 0 (in ℋ) as x,y → 0 in 𝒢. The basic question here is whether ω is approximable by a true homomorphism a in the sense that ω(x)-a(x) → 0 in ℋ as x → 0 in 𝒢. Our main result is that quasi-homomorphisms ω:𝒢 → ℋ are approximable in the following two cases: ∙ 𝒢 is a product of locally compact abelian groups and ℋ is either ℝ or the circle group 𝕋. ∙ 𝒢 is either ℝ or 𝕋 and ℋ is a Banach space. This is proved by adapting a classical procedure in the theory of twisted sums of Banach spaces. As an application, we show that every abelian extension of a quasi-Banach space by a Banach space is a topological vector space. This implies that most classical quasi-Banach spaces have only approximable (real-valued) quasi-additive functions. },
author = {Félix Cabello Sánchez},
journal = {Fundamenta Mathematicae},
keywords = {quasi-homomorphisms; Cauchy's equation; quasi-additive mappings; stability theory; homology},
language = {eng},
number = {3},
pages = {255-270},
title = {Quasi-homomorphisms},
url = {http://eudml.org/doc/283225},
volume = {178},
year = {2003},
}

TY - JOUR
AU - Félix Cabello Sánchez
TI - Quasi-homomorphisms
JO - Fundamenta Mathematicae
PY - 2003
VL - 178
IS - 3
SP - 255
EP - 270
AB - We study the stability of homomorphisms between topological (abelian) groups. Inspired by the "singular" case in the stability of Cauchy's equation and the technique of quasi-linear maps we introduce quasi-homomorphisms between topological groups, that is, maps ω: 𝒢 → ℋ such that ω(0) = 0 and ω(x+y) - ω(x) - ω(y) → 0 (in ℋ) as x,y → 0 in 𝒢. The basic question here is whether ω is approximable by a true homomorphism a in the sense that ω(x)-a(x) → 0 in ℋ as x → 0 in 𝒢. Our main result is that quasi-homomorphisms ω:𝒢 → ℋ are approximable in the following two cases: ∙ 𝒢 is a product of locally compact abelian groups and ℋ is either ℝ or the circle group 𝕋. ∙ 𝒢 is either ℝ or 𝕋 and ℋ is a Banach space. This is proved by adapting a classical procedure in the theory of twisted sums of Banach spaces. As an application, we show that every abelian extension of a quasi-Banach space by a Banach space is a topological vector space. This implies that most classical quasi-Banach spaces have only approximable (real-valued) quasi-additive functions.
LA - eng
KW - quasi-homomorphisms; Cauchy's equation; quasi-additive mappings; stability theory; homology
UR - http://eudml.org/doc/283225
ER -