Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions

Abstract

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I1, and Lm:A→A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕm:A→A and an additive higher map ζm:A→Z(A), which annihilates every (n−1)th commutator pn(S1,S2,⋯,Sn) with S1S2=0 such that Lm(S)=ϕm(S)+ζm(S)forallS∈A. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results.