Nonlinear Mixed Lie Triple Derivations by Local Actions on Von Neumann Algebras

Abstract

As a generalization of global mappings, we study a class of non-global mappings in this note. Let $\mathcal{A} ⊆ B(\mathcal{H})$ be a von Neumann algebra without abelian direct summands. We prove that if a map $δ:\mathcal{A}→\mathcal{A}$ satisfies $δ([[A,B]_∗,C]) = [[δ(A),B]_∗,C]+ [[A,δ(B)]_∗,C]+[[A,B]_∗,δ(C)]$ for any $A,B,C ∈ \mathcal{A}$ with $A^∗B^∗C =0,$ then $δ$ is an additive ∗-derivation. As applications, our results are applied to factor von Neumann algebras, standard operator algebras, prime ∗-algebras and so on.