Korean J. Math. Vol. 27 No. 4 (2019) pp.969-976
DOI: https://doi.org/10.11568/kjm.2019.27.4.969

Nonlinear $\xi$-Lie-$\ast$-derivations on von Neumann algebras

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Published: Dec 30, 2019
Keywords:
Linear preserver, Nest subalgebra, Von Neumann algebra.
Mathematics Subject Classification:
47B47, 46l40

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Aili Yang
College of Science Xi’an University of Science and Technology Xi’an 710054, P. R. China

Abstract

Let $\mathscr{B(H)}$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathscr{H}$ and $\mathscr{M}\subseteq\mathscr{B(H)}$ be a von Neumann algebra without central abelian projections. Let $\xi$ be a non-zero scalar. In this paper, it is proved that a mapping $\varphi:\mathscr{M}\rightarrow\mathscr{B(H)}$ satisfies $\varphi([A,B]^{\xi}_{\ast})=[\varphi(A),B]^{\xi}_{\ast}+[A,\varphi(B)]^{\xi}_{\ast}$ for all $A,B\in\mathscr{M}$ if and only if $\varphi$ is an additive $\ast$-derivation and $\varphi(\xi A)=\xi\varphi(A)$ for all $A\in\mathscr{M}.$



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College of Science Xi’an University of Science and Technology Xi’an 710054 P. R. China

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