Lie Triple Derivations on von Neumann Algebras

Abstract

Let $\mathcal{A}$ be a von Neumann algebra with no central abelian projections. It is proved that if an additive map δ : $\mathcal{A}$ → $\mathcal{A}$ satisfies δ([[a, b], c]) = [[δ(a), b], c]+[[a, δ(b)], c]+ [[a, b], δ(c)] for any a, b, c ∈ $\mathcal{A}$ with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection in $\mathcal{A}$), then there exist an additive derivation d from $\mathcal{A}$ into itself and an additive map f : $\mathcal{A}$ → $\mathcal{Z}_\mathcal{A}$ vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that δ(a) = d(a) + f(a) for any a ∈ $\mathcal{A}$.