Local and 2-Local Lie Derivations of Operator Algebras on Banach Spaces

Abstract

Let X be a Banach space of dimension > 2. We show that every local Lie derivation of B(X) is a Lie derivation, and that a map of B(X) is a 2-local Lie derivation if and only if it has the form \({A \mapsto AT - TA + \psi(A)}\), where \({T \in B(X)}\) and ψ is a homogeneous map from B(X) into \({\mathbb{F}I}\) satisfying \({\psi(A + B) = \psi(A)}\) for \({A, B \in B(X)}\) with B being a sum of commutators.