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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 $A\in\mathscr{M}.$
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참고문헌 (9)
참고문헌 신청
B.E. Johnson / 1996 / Symmetric amenability and the nonexistence of Lie and Jordan derivations / Math. Proc. Cambridge Philos. Soc 120:455~473
M. Mathieu / 2003 / The structure of Lie derivations on Cꠗ-algebras / J. Funct. Anal 202:504~525
J. -H. Zhang / 2003 / Lie derivations on nest subalgebras of von Neumann algebras / Acta Math. Sinica 46:657~664
W. S. Cheung / 2003 / Lie derivation of triangular algebras / Linear and Multilinear Al-gebra 51:299~310
X.F. Qi / 2009 / Additive Lie (-Lie) derivations and generalized Lie (-Lie) derivations on nest algebras / Linear Algebra Appl 431:843~854
M. Mathieu / 2003 / The structure of Lie derivations on Cꠗ-algebras / J. Funct. Anal 202:504~525
J. -H. Zhang / 2003 / Lie derivations on nest subalgebras of von Neumann algebras / Acta Math. Sinica 46:657~664
W. S. Cheung / 2003 / Lie derivation of triangular algebras / Linear and Multilinear Al-gebra 51:299~310
X.F. Qi / 2009 / Additive Lie (-Lie) derivations and generalized Lie (-Lie) derivations on nest algebras / Linear Algebra Appl 431:843~854
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