Characterizations of Lie derivations of factor von Neumann algebras

Abstract

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).