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 map-pings in this note. Let $\mathcal{A} \subseteq B(\mathcal{H})$ be a von Neumann algebra without abelian direct summands. We prove that if a map δ : A → A satisfies δ([[A,B]∗,C]) = [[δ(A),B]∗,C]+[[A,δ(B)]∗,C]+[[A,B]∗,δ(C)] for any A,B,C ∈ 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.

Keywords

Nonlinear mixed Lie triple derivation / ∗-derivation / von Neumann algebra