2-local S mappings with respect to τ on some *-subalgebras of

Abstract

We prove that if Δ is a norm-continuous weak ∗ -2-local derivation on a von Neumann algebra M and satisfies Δ ( p + iμq ) = Δ ( p ) + iμ Δ ( q ) for every pair of projections p and q in M , and every μ ∈ R , then Δ is a derivation. We further show that every weak-local derivation of an R ∗ -algebra is a derivation and that every weak-2-local inner derivation on a unital R ∗ -algebra is a derivation. Finally, we show that if ϕ is a 2-local Lie ∗ -isomorphism of a factor M with no central summands of type I 1 or I 2 , then ϕ only has one of two forms: (1) θ + λ where θ is a ∗ -isomorphism and λ is a ∗ -linear map from M into C I M which annihilates every commutator of M or (2) - θ + λ where θ is a ∗ -anti-isomorphism and λ as before.