Document Type : Research Paper

Author

• N. Ghobadipour

Department of Mathematics, Urmia University, Urmia, Iran.

Abstract

A unital $C^{\ast }$-algebra $\mathcal{A}$ endowed with the Lie product $[x,y]=xy-yx$ on $\mathcal{A}$ is called a Lie $C^{\ast }$-algebra. Let $\mathcal{A}$ be a Lie $C^{\ast }$-algebra and $g,h:\mathcal{A}\to \mathcal{A}$ be $\mathbb{C}$-linear mappings. A $\mathbb{C}$-linear mapping $f:\mathcal{A}\to \mathcal{A}$ is called a Lie $(g,h)$--double derivation if $f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all $a,b\in \mathcal{A}$. In this paper, our main purpose is to prove the generalized Hyers–Ulam–Rassias stability  of Lie $\ast$-double derivations on Lie $C^{\ast }$-algebras associated with the
following additive mapping:
$$\sum _{k=2}^n(\sum _{i_1=2}^k\sum _{i_2=i_1+1}^{k+1}...\sum _{i_{n-k+1}=i_{n-k}+1}^n)f(\sum _{i=1,i\ne i_1,..,i_{n-k+1}}^n{x}_i-\sum _{r=1}^{n-k+1}{x}_{i_r})+f(\sum _{i=1}^n{x}_i)=2^{n-1}f(x_1)$$
 for a fixed positive integer $n$ with $n\ge 2.$

Keywords

• Generalized Hyers-Ulam-Rassias stability
• $\ast$-double derivation
• Lie $C^{\ast }$-algebra