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Quasi-homomorphisms. (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 $\omega :𝒢\to ℋ$ such that $\omega \left(0\right)=0$ and

$\omega \left(x+y\right)-\omega \left(x\right)-\omega \left(y\right)\to 0$

(in $ℋ$) as $x,y\to 0$ in $𝒢$. 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 $ℋ$ as $x\to 0$ in $𝒢$. Our main result is that quasi-homomorphisms $\omega :𝒢\to ℋ$ 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.”

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