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

Abstract

Petr Novotný and Jiřĺ Hrivnák \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 $C^{\ast }$-algebras associated with the following functional equation $$f(\frac{x_2-x_1}{3})+f(\frac{x_1-3x_3}{3})+f(\frac{3x_1+3x_3-x_2}{3})=f(x_1).$$ }

10 pages