von Neumann代数上的可导映射与导子

Characterization of Derivations on von Neumann Algebras by Derivable Maps

设A为von Neumann代数,Ω∈A为任意但固定的算子。本文证明有界线性映射δ:A→A在Ω可导,即δ(AB)=δ(A)B+Aδ(B),A,B∈A,AB=Ω当且仅当存在导子τ:A→A使得δ(A)=τ(A)+δ(Ι)A,∀A∈A,其中δ(Ι)∈Z(A)且δ(Ι)Ω=0。特别地,若A是没有Ι1型直和项的von Neumann代数或真无限von Neumann代数,则将线性且连续的假设弱化为可加仍得到上述结果。

Let be a von Neumann algebra and Ω∈A be an arbitrary but fixed operator. In this paper, we show that a linear bounded map δ:A→A is derivable at Ω, that is, δ(AB)=δ(A)B+Aδ(B) for every A,B∈A with AB=Ω if and only if there exists a derivation τ:A→A such that δ(A)=τ(A)+δ(Ι)A for all A∈A where δ(Ι) is in the center of A and δ(Ι)Ω=0. In particular, if A is a von Neumann algebra with no summands of type Ι1 or a properly infinite von Neumann algebra, similar results can be obtained by weakening the linearity and continuity assumption of δ into additivity.