Lie ^*-double derivations on Lie C^*-algebras

A unital C^*-algebra \mathcal{A} endowed with the Lie product [x,y]=xy- yx on \mathcal{A} is called a Lie C^*-algebra. Let \mathcal{A} be a Lie C^*-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 *-double derivations on Lie C^*-algebras associated with thefollowing additive mapping:\sum^{n}_{k=۲}(\sum^{k}_{i_{۱}=۲} \sum^{k+۱}_{i_{۲}=i_{۱}+۱}...\sum^{n}_{i_{n-k+۱}=i_{n-k}+۱}) f(\sum^{n}_{i=۱, i\neqi_{۱},..,i_{n-k+۱} } x_{i}-\sum^{n-k+۱}_{ r=۱}x_{i_{r}})+f(\sum^{n}_{ i=۱} x_{i})=۲^{n-۱} f(x_{۱}) for a fixed positive integer n with n \geq ۲.

Keywords:

Generalized Hyers-Ulam-Rassias stability , *-double derivation , Lie C^*-algebra