The second nonlinear mixed bi-skew lie triple derivations on factor von Neumann algebras

Abstract

Let M be a factor von Neumann algebra with dim M> 1. In this paper, it is shown that if a map ϕ : M → M satisfies ϕ([[A,B],C]⋄) = [[ϕ(A),B],C]⋄ + [[A, ϕ(B)],C]⋄+[[A,B], ϕ(C)]⋄ for all A,B,C ∈M, then ϕ is an additive ∗-derivation, where [A,B]⋄ = AB∗ − BA∗.