Characterizations of Lie derivations of factor von Neumann algebras

摘要

Let R be a factor von Neumann algebra with dimension > 4. In this article, we prove that if δ: R → R is a linear map satisfying for any x, y ∈ R with xy = 0 (resp., xy = p, where p is a fixed nontrivial projection of R), then δ = d + τ, where d is a derivation of R and τ: R → CI (where C is the field of complex numbers) is a linear map vanishing at commutators [x, y] with xy = 0 (resp., xy = p).