Quasi-homomorphisms

Félix Cabello Sánchez

Fundamenta Mathematicae, Tome 178 (2003) no. 3, p. 255.

Résumé

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.

Détail

Classification : 39B52, 39B82
Mots-clés : quasi-homomorphisms, Cauchy's equation, quasi-additive mappings, stability theory, homology