The Second Nonlinear Mixed Lie Triple Derivations on Finite Von Neumann Algebras

Let M be a finite von Neumann algebra with no central summands of type I1. Suppose that L : M → Mis the second nonlinear mixed Lie triple derivation. Then L is an additive ∗-derivation. We also show that each local and 2-local second Lie triple derivation on finite von Neumann algebras with no central summands of type I1 is a ∗-derivation. Besides, each local and 2-local second Lie triple derivation on factor von Neumann algebras M with dim M 1 is also a ∗-derivation.