Lie Triple Derivations on Factor Von Neumann Algebras

Let <TEX>A</TEX> be a factor von Neumann algebra with dimension greater than 1. We prove that if a linear map <TEX>δ:A→A</TEX> satisfies <TEX>δ([[a,b],c])=[[δ(a),b],c]+[[a,δ(b),c]+[[a,b],δ(c)]</TEX> for any <TEX>a,b,c∈A</TEX> with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection of <TEX>A</TEX>), then there exist an operator <TEX>T∈A</TEX> and a linear map <TEX>f:A→CI</TEX> vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that <TEX>δ(a)=aT-Ta+f(a)</TEX> for any <TEX>a∈A</TEX>.