Characterization of Lie higher Derivations on $C^{*}$-algebras
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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 $ngeq0$.
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Journal title
volume 41 issue 4
pages 901- 906
publication date 2015-08-01
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