Non-global Multiplicative Lie Triple Derivations on Rings

Abstract

Let $\mathfrak {R}$ be a ring containing a nontrivial idempotent with center $\mathcal {Z}(\mathfrak {R})$. In the present article, it is shown that under certain restrictions every map $\xi :\mathfrak {R}\rightarrow \mathfrak {R}$ (not necessarily additive) satisfying $\xi ([[S, T], U])=[[\xi (S), T], U]+[[S,\xi (T)],$ $U]+[[S, T],\xi (U)]$ for all $S, T, U\in \mathfrak {R}$ with $STU=0,$ is almost additive, that is, $\xi (S+T)-\xi (S)-\xi (T)\in \mathcal {Z}(\mathfrak {R}).$ In addition, if $\mathfrak {R}$ is a 2-torsion free prime ring, then $\xi$ is of the form $\xi =\partial +\eta ,$ where $\partial$ is a derivation from $\mathfrak {R}$ into its central closure $\mathfrak {S}$ and $\eta$ is a map from $\mathfrak {R}$ into its extended centroid $\mathfrak {C}$ such that $\eta (S+T)-\eta (S)-\eta (T)\in \mathcal {Z}(\mathfrak {R})$ and $\eta ([[S, T], U])=0$ for all $S,T,U\in \mathfrak {R}$ with $STU=0.$ The obtained results are then applied to standard operator algebras, factor von Neumann algebras and the algebra of all bounded linear operators.