The Characterization of 2-Local Lie Automorphisms of Some Operator Algebras

Abstract

Let M ⊑ B(X) be an algebra with nontrivial idempotents or nontrivial projections if M is a *-algebra and ZM = ℂI. In this paper, the notion of (strong) 2-local Lie automorphism normalized property is introduced and it is proved that if M has 2-local Lie automorphism normalized property and Φ: M → M is an almost additive surjective 2-local Lie isomorphism with idempotent decomposition property, then → = Ψ + τ, where Ψ is an automorphism of M or the negative of an anti-automorphism of M and τ is a homogenous map from M into ℂI. Moreover, it is proved that nest algebras on a separable complex Hilbert space II with dimII >2 and factor von Neumann algebras on a separable complex Hilbert space H with dimH > 2 have strong 2-local Lie automorphism normalized property.