Abstract:

Let ${\cal A}$ be a von Neumann algebra with no central abelian projections, $P\in{\cal A}$ be a projection with $\underline{P}=0$ and $\overline{P}=I$. An additive map $\delta:{\cal A}\rightarrow{\cal A}$ is said to be Lie derivable at $\Omega\in{\cal A}$, if $\delta([A, B])=[\delta(A), B]+[A, \delta(B)]$ for any $A, B\in{\cal A}$ with $AB=\Omega.$ We show that, if $\Omega\in{\cal A}$ such that $P\Omega=\Omega$, then $\delta$ is Lie derivable at $\Omega$ if and only if there exist a derivation $\tau:{\cal A} \rightarrow {\cal A}$ and and additive map $f: {\cal A}\rightarrow {\cal Z}({\cal A})$ vanishing at commutators $[A, B]$ with $AB=\Omega$ such that $\delta(A)=d(A)+f(A), \forall A\in {\cal A}.$ In particular, if ${\cal A}$ is a factor von Neuamnn algebra and $\Omega\in {\cal A}$ such that $\mbox{ker}(\Omega)\neq {0}$ or $\overline{\mbox{ran}(\Omega)}\neq H, $ then $\delta$ is Lie derivable at $\Omega$ if and only if it has the above form.

Key words: von Neumann algebras, Lie derivations, Lie derivable maps, Central carrier