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

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 $n\geqslant0$‎. ‎We show that‎ ‎$\{L_{n}\}_{n=0}^{\infty}$ is a Lie higher derivation if and only if‎ ‎there exist a higher derivation‎ ‎$\{D_{n}:\mathcal{A}\rightarrow\mathcal{A}\}_{n=0}^{\infty}$ 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 ‎$x,y\in\mathcal{A}$ and all $n\geq0$‎.