Characterization of Lie higher Derivations on C ∗ -algebras

Let A be a C ∗ -algebra and Z(A) the‎ ‎center of A ‎. ‎A sequence {L n} ∞ n=0 of‎ ‎linear mappings on A with L 0 =I ‎, ‎where I is the‎ ‎identity mapping‎ ‎on A ‎, ‎is called a Lie higher derivation if‎ ‎L n [x,y]=∑ i+j=n [L i x,L j y] for all x,y∈‎A and all n⩾0 ‎. ‎We show that‎ ‎{L n} ∞ n=0 is a Lie higher derivation if and only if‎ ‎there exist a higher derivation‎ ‎{D n: A→A} ∞ n=0 and a‎ ‎sequence of linear mappings {Δ n: A→‎‎Z(A)} ∞ n=0 ‎such that Δ 0 =0 ‎, ‎Δ n ([x,y])=0 and L n =D n +Δ n for every‎ ‎x,y∈A and all n≥0 ‎.