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

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)+iμΔ(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 I1 or I2, 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 CIM which annihilates every commutator of M or (2) -θ+λ where θ is a ∗-anti-isomorphism and λ as before.