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


Department of Mathematics, Urmia University, Urmia, Iran.


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:
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})
 for a fixed positive integer $n$ with $n geq 2.$