Nonlinear $\xi$-Lie-$\ast$-derivations on von Neumann algebras
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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 $A\in\mathscr{M}.$
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 $A\in\mathscr{M}.$
참고문헌 (9)
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