The characterization of 2-local Lie automorphisms of some operator algebras

Let M ⊆ B ( X ) be an algebra with nontrivial idempotents or nontrivial projections if M is a ∗ -algebra and Z M = C I . 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 Φ = 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 C I . Moreover, it is proved that nest algebras on a separable complex Hilbert space H with dim H > 2 and factor von Neumann algebras on a separable complex Hilbert space H with dim H ≥ 2 have strong 2-local Lie automorphism normalized property.