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LIE TRIPLE DERIVATIONS ON FACTOR VON NEUMANN ALGEBRAS

  • Liu, Lei
  • Received : 2014.03.19
  • Published : 2015.03.31

Abstract

Let $\mathcal{A}$ be a factor von Neumann algebra with dimension greater than 1. We prove that if a linear map ${\delta}:\mathcal{A}{\rightarrow}\mathcal{A}$ satisfies $${\delta}([[a,b],c])=[[{\delta}(a),b],c]+[[a,{\delta}(b),c]+[[a,b],{\delta}(c)]$$ for any $a,b,c{\in}\mathcal{A}$ with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection of $\mathcal{A}$), then there exist an operator $T{\in}\mathcal{A}$ and a linear map $f:\mathcal{A}{\rightarrow}\mathbb{C}I$ vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that ${\delta}(a)=aT-Ta+f(a)$ for any $a{\in}\mathcal{A}$.

Keywords

Lie derivations;Lie triple derivations;factor von Neumann algebras

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Cited by

  1. LIE -HIGHER DERIVATIONS AND LIE -HIGHER DERIVABLE MAPPINGS vol.96, pp.02, 2017, https://doi.org/10.1017/S0004972717000338

Acknowledgement

Supported by : NNSF of China