Additive Lie ($\xi$-Lie) Derivations and Generalized Lie ($\xi$-Lie) Derivations on Prime Algebras

TL;DR

This paper characterizes additive (generalized) $\xi$-Lie derivations on prime algebras and operator algebras, showing their structure and conditions under which they are equivalent to derivations or sums involving derivations.

Contribution

It provides a complete characterization of additive (generalized) $\xi$-Lie derivations on prime and operator algebras, extending previous results to new algebraic structures.

Findings

Additive (generalized) $\xi$-Lie derivations are characterized as sums of derivations and central maps.

For $\xi eq 1$, such derivations are equivalent to derivations satisfying a specific scalar condition.

Results apply to operator algebras like Banach space and von Neumann algebras.

Abstract

The additive (generalized) $\xi$-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumption, that an additive map $L$ is an additive (generalized) Lie derivation if and only if it is the sum of an additive (generalized) derivation and an additive map from the algebra into its center vanishing all commutators; is an additive (generalized) $\xi$-Lie derivation with $\xi\not=1$ if and only if it is an additive (generalized) derivation satisfying $L(\xi A)=\xi L(A)$ for all $A$. These results are then used to characterize additive (generalized) $\xi$-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.