CSL代数上在单位元Ⅰ点Jordan高阶可导的映射

Jordan Higher Derivable Maps at Ⅰ on CSL Algebras

设L是希尔伯特空间H上的一个CSL,AlgL是相应地CSL代数。一族线性映射δ_={δ_n,δ_n:AlgL→AlgL,n∈N}在Ω∈AlgL Jordan高阶可导,如果对所有n∈N,∑[δ_i(A)δ_j(B)+δ_j(B)δ_i(A)]=δ_(Ω),其中A,B∈AlgL,AB+BA=Ω。本文给出了一族线性映射δ_={δ_n:AlgL→AlgL}在单位元I点Jordan高阶可导的充要条件。利用此结果证明了不可约CDCSL代数,因子von Neumann代数上的套子代数(特别地,希尔伯特空间套代数)到其自身的一族线性映射δ_={δ_n,n∈N}在I点Jordan高阶可导当且仅当它是一个高阶导子。

Let L be a CSL on a Hilbert space H and AlgL be the associated CSL algebras. We say that a family of linear maps δ_ ={δ_n,δ_n:AlgL → AlgL,n ∈ N} is Jordan higher derivable at Ω ∈ AlgL if∑i + j = n[δ_]i(A)δ_j(B)+ δ_j(B)δ_i(A) = δ_(Ω) for all n ∈ N, A,B ∈ AlgL with AB + BA = Ω. In this paper, we give a necessary and sufficient condition for a family of linear maps δ_ ={δ_n,AlgL → AlgL,n ∈ N} to be Jordan derivable at I. As its application, we show that a family of linear maps δ_ ={δ_n,n ∈ N} from an irreducible CDCSL algebra or a nest subalgebra of a factor von Nuemann algebra(in particular, a Hilbert space nest algebra)into itself, which is Jordan higher derivable at I is a higher derivation.