(untitled) (Journal of Mathematical Study, 2024)
As a generalization of global mappings, we study a class of non-global mappings in this note. Let A ⊆ B(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.