*(English)*Zbl 1051.39032

Author’s 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 $\omega :\mathcal{G}\to \mathscr{H}$ such that $\omega \left(0\right)=0$ and

(in $\mathscr{H}$) as $x,y\to 0$ in $\mathcal{G}$. The basic question here is whether $\omega $ is approximable by a true homomorphism $a$ in the sense that $\omega \left(x\right)-a\left(x\right)\to 0$ in $\mathscr{H}$ as $x\to 0$ in $\mathcal{G}$. Our main result is that quasi-homomorphisms $\omega :\mathcal{G}\to \mathscr{H}$ are approximable in the following two cases:

$\u2022$ $\mathcal{G}$ is a product of locally compact abelian groups and $\mathscr{H}$ is either $\mathbb{R}$ or the circle group $\mathbb{T}$.

$\u2022$ $\mathcal{G}$ is either $\mathbb{R}$ or $\mathbb{T}$ and $\mathscr{H}$ 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.”

A reference for both main concepts and results in the subject is the book of *D. H. Hyers, G. Isac* and *Th. M. Rassias* [Stability of functional equations in several variables (Progress in Nonlinear Differential Equations and their Applications 34, Boston, Birkhäuser) (1998; Zbl 0907.39025)].

##### MSC:

39B82 | Stability, separation, extension, and related topics |

39B52 | Functional equations for functions with more general domains and/or ranges |