Approximately 2Local Derivations on the Semi-finite von Neumann Algebras

Abstract:

The definition of approximately 2-local derivation on von Neumann algebras is introduced based on the definitions of approximately local derivation and 2-local derivation. Approximately 2-local derivations on semi-finite von Neumann algebras are studied. Let M be a von Neumann algebra and Δ: M→M be an approximately 2-local derivation. It is easy to obtain that Δ is homogeneous and Δ satisfies Δ(x2) =Δ(x)x+xΔ(x) for any x∈M. Besides, if M is a von Neumann algebra with a faithful normal semi-finite trace τ, then a sufficient condition for Δ to be additive is given, that is, Δ(Mτ)⊆Mτ, where Mτ={x∈M:τ(|x|)<∞}. In all, if Δ is an approximately 2-local derivation on a semi-finite von Neumann algebra with a faithful normal semi-finite trace τ and satisfies Δ(Mτ) ⊆Mτ, where Mτ={x∈M:τ(|x|)<∞}, by the conclusion that the Jordon derivation from a 2-torsion free semi-prime ring to itself is a derivation, it follows that Δ is a derivation.