Characterization of Multiplicative Lie Triple Derivations on Rings

Xiaofei Qi

doi:10.1155/2014/739730

Abstract

LetRbe a ring having unit 1. Denote byZRthe center ofR. Assume that the characteristic ofRis not 2 and there is an idempotent elemente∈Rsuch thataRe=0⇒a=0 and aR1-e=0⇒a=0. It is shown that, under some mild conditions, a mapL:R→Ris a multiplicative Lie triple derivation if and only ifLx=δx+hxfor allx∈R, whereδ:R→Ris an additive derivation andh:R→ZRis a map satisfyingha,b,c=0for alla,b,c∈R. As applications, all Lie (triple) derivations on prime rings and von Neumann algebras are characterized, which generalize some known results.

Highlights

Let R be an associative ring with the center Z(R)
Bresǎ r in [1] proved that every additive Lie derivation on a prime ring R with characteristic not 2 can be decomposed as τ + ζ, where τ is an additive derivation from R into its central closure and ζ is an additive map of R into the extended centroid C sending commutators to zero
Mathieu and Villena [2] showed that every linear Lie derivation on a C∗-algebra is standard, that is, can be decomposed as the form τ + h, where τ is a derivation and h is a central valued linear map vanishing at each commutator

Summary

Introduction

Let R be an associative ring with the center Z(R). For any element a, b ∈ R, we set [a, b] = ab − ba. Some characterizations of multiplicative (additive) Lie (triple) derivations on prime rings and von Neumann algebras are obtained, respectively (Corollaries 3–7). (2) There exist an additive derivation δ : M → M and a map h : M → Z(M) vanishing at each Lie triple product [[A, B], C] such that L(A) = δ(A) + h(A), for all A ∈ M.

Objectives

Results

Conclusion