Quasi-homomorphisms
Volume 178 / 2003
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 $\omega:{\cal G}\to{\cal H}$ 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.
