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 I_1 . Suppose that L: ℳ→ℳ is 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 I_1 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.