Characterization of Lie higher Derivations on $C^{*}$-algebras

Abstract

Let $mathcal{A}$ be a $C^*$-algebra and $Z(mathcal{A})$ the‎ ‎center of $mathcal{A}$‎. ‎A sequence ${L_{n}}_{n=0}^{infty}$ of‎ ‎linear mappings on $mathcal{A}$ with $L_{0}=I$‎, ‎where $I$ is the‎ ‎identity mapping‎ ‎on $mathcal{A}$‎, ‎is called a Lie higher derivation if‎ ‎$L_{n}[x,y]=sum_{i+j=n} [L_{i}x,L_{j}y]$ for all $x,y in ‎mathcal{A}$ and all $ngeqslant0$‎. ‎We show that‎ ‎${L_{n}}_{n=0}^{infty}$ is a Lie higher derivation if and only if‎ ‎there exist a higher derivation‎ ‎${D_{n}:mathcal{A}rightarrowmathcal{A}}_{n=0}^{infty}$ and a‎ ‎sequence of linear mappings ${Delta_{n}:mathcal{A}rightarrow‎ ‎Z(mathcal{A})}_{n=0}^{infty}$‎ ‎such that $Delta_0=0$‎, ‎$Delta_n([x,y])=0$ and $L_n=D_n+Delta_n$ for every‎ ‎$x,yinmathcal{A}$ and all $ngeq0$‎.