半有限von Neumann代数上的逼近2-局部导子

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

摘要:

在逼近局部导子和2-局部导子的基础上, 给出了von Neumann代数上逼近2-局部导子的定义. 研究了半有限von Neumann代数上的逼近2-局部导子. 设M是一个von Neumann代数,Δ: M→M 是一个逼近2-局部导子. 证明Δ具有齐次性并且满足对于任意的x∈M有Δ(x2)=Δ(x)x+xΔ(x). 若M是具有半有限迹τ的von Neumann代数, 给出了M到其自身的逼近2-局部导子Δ具有可加性的一个充分条件, 即Δ满足Δ(Mτ)?Mτ, 其中Mτ={x∈M:τ(|x|)<∞}. 从而由2-torsion free半素环R 到R自身的Jordon导子是一个导子得知, 具有半有限迹τ的von Neumann代数M到其自身的逼近2-局部导子Δ若满足Δ (Mτ) ⊆Mτ, 其中Mτ={x∈M:τ(|x|)<∞}, 则Δ是一个导子.

关键词:逼近2-局部导子; 半有限von Neumann 代数; 导子

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.

Key words:approximately 2-local derivation; semi-finite von Neumann algebra; derivation