Stability of $(\alpha,\beta,\gamma)-$derivations on Lie $C^*-$algebras

TL;DR

This paper studies the stability of a generalized class of derivations called $(eta,eta,eta)$-derivations on Lie $C^*$-algebras, focusing on solutions to a specific functional equation and their approximate solutions.

Contribution

It extends the stability analysis of $(eta,eta,eta)$-derivations to Lie $C^*$-algebras, providing new insights into their structure and stability properties.

Findings

Established stability conditions for $(eta,eta,eta)$-derivations.

Connected the functional equation to derivation stability.

Provided conditions under which approximate solutions are near true derivations.

Abstract

Petr Novotn\'y and Ji\v{r}\'l Hrivn\'ak \cite{Nov} investigated generalize the concept of Lie derivations via certain complex parameters and obtained various Lie and Jordan operator algebras as well as two one- parametric sets of linear operators. Moreover, they established the structure and properties of $(\alpha,\beta,\gamma)-$derivations of Lie algebras. We say a functional equation $(\xi)$ is stable if any function $g$ satisfying the equation $(\xi)$ {\it approximately} is near to true solution of $(\xi).$ In the present paper, we investigate the stability of $(\alpha,\beta,\gamma)-$derivations on Lie $C^*$-algebras associated with the following functional equation $$f(\frac{x_2-x_1}{3})+f(\frac{x_1-3 x_3}{3})+ f(\frac{3x_1+3x_3-x_2}{3})=f(x_1).$$ }