Quasi-homomorphisms

Félix Cabello Sánchez¹

¹ Departamento de Matemáticas Universidad de Extremadura Avenida de Elvas 06071 Badajoz, Spain

Fundamenta Mathematicae, Tome 178 (2003) no. 3, pp. 255-270

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 $\omega:{\cal G}\to{\cal H}$ such that $\omega(0)=0$ and $$ \omega(x+y)-\omega(x)-\omega(y)\to 0 $$ (in ${\cal H}$) as $x,y\to 0$ in ${\cal G}$. The basic question here is whether $\omega$ is approximable by a true homomorphism $a$ in the sense that $\omega(x)-a(x)\to 0$ in ${\cal H}$ as $x\to 0$ in ${\cal G}$. Our main result is that quasi-homomorphisms $\omega:{\cal G}\to{\cal H}$ are approximable in the following two cases:$\bullet$ ${\cal G}$ is a product of locally compact abelian groups and ${\cal H}$ is either $\mathbb R$ or the circle group $\mathbb T$. $\bullet$ ${\cal G}$ is either $\mathbb R$ or $\mathbb T$ and ${\cal H}$ 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

DOI : 10.4064/fm178-3-5

Mots-clés : study stability homomorphisms between topological abelian groups inspired singular stability cauchys equation technique quasi linear maps introduce quasi homomorphisms between topological groups maps omega cal cal omega omega omega omega cal cal basic question here whether omega approximable homomorphism sense omega a cal cal main result quasi homomorphisms omega cal cal approximable following cases bullet cal product locally compact abelian groups cal either mathbb circle group mathbb bullet cal either mathbb mathbb cal banach space proved adapting classical procedure theory twisted sums banach spaces application every abelian extension quasi banach space banach space topological vector space implies classical quasi banach spaces have only approximable real valued quasi additive functions