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

Authors

  • A. R. Janfada‎ Department of‎ ‎Science‎, ‎University of Birjand‎, ‎P.O‎. ‎Box 414‎, ‎Birjand 9717851367‎, ‎Birjand‎, ‎Iran
  • H. Saidi Department of Science‎, ‎University‎ ‎of Birjand‎, ‎P.O‎. ‎Box 414‎, ‎Birjand 9717851367‎, ‎Birjand‎, ‎Iran
  • M. Mirzavaziri Department of‎ ‎Pure Mathematics‎, ‎Ferdowsi University of Mashhad‎, ‎P.O‎. ‎Box 1159‎, ‎Mashhad 91775‎, ‎Mashhad‎, ‎Iran
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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