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

Let M subset of BoXTHORN be an algebra with nontrivial idempotents or nontrivial projections if M is a *-algebra and Z(M) = 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 Phi : M -> M is an almost additive surjective 2-local Lie isomorphism with idempotent decomposition property, then Phi = psi + tau, where W is an automorphism of M or the negative of an anti-automorphism of M and tau is a homogenous map from M into CI. Moreover, it is proved that nest algebras on a separable complex Hilbert space H with dimH > 2 and factor von Neumann algebras on a separable complex Hilbert space H with dimH >= 2 have strong 2-local Lie automorphism normalized property.