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 $bbb c$ -- linear mappings. a $bbb c$ -- 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 the following additive mapping: begin{align*} sum^{n}_{k=2}(sum^{k}_{i_{1}=2} sum^{k+1}_{i_{2}=i_{1}+1}... sum^{n}_{i_{n-k+1}=i_{n-k}+1}) f( sum^{n}_{i=1, ineq i_{1},..,i_{n-k+1} } x_{i}&-sum^{n-k+1}_{ r=1}x_{i_{r}})+f(sum^{n}_{ i=1} x_{i}) &=2^{n-1} f(x_{1}) end{align*} for a fixed positive integer $n$ with $n geq 2.$