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

Abstract

Let $${\mathcal {M}}$$ be a finite von Neumann algebra with no central summands of type $${I}_{1}$$ . Suppose that $$L: {\mathcal {M}}\rightarrow {\mathcal {M}}$$ 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 $${\mathcal {M}}$$ with dim $${\mathcal {M}}>1$$ is also a $$*$$ -derivation.