因子von Neumann代数上的非全局非线性Lie三重可导映射

Non-global Nonlinear Lie Triple Derivable Maps on Factor von Nuemann Algebras

设M是Hilbert空间H上维数大于1的因子vonNeumann代数,用代数分解方法证明了:如果非线性映射δ:M→M满足对任意的A,B,C∈M且ABC=0,有δ([[A,B],C])=[[δ(A),B],C]+[[A,δ(B)],C]+[[A,B],δ(C)],则存在可加导子d:M→M,使得对任意的A∈M,有δ(A)=d(A)+τ(A)I,其中τ:CI是一个非线性映射,满足对任意的A,B,C∈M且ABC=0时,有τ([[A,B],C])=0。

Let M be a factor von Neumann algebra with dimension greater than 1 on a Hilbert space H . With the help of algebraic decomposition method,we proved that if a nonlinear map δ: M → M satisfied δ([[A,B],C])=[[δ(A),B],C]+[[A,δ(B)],C]+[[A,B],δ(C)] for any A,B,C∈ M with ABC =0,then there existed an additive derivation d: M → M , such that δ(A)=d(A)+τ(A) I for any A∈ M ,where τ: M CI is a nonlinear map such that τ([[A,B],C])=0 with ABC =0 for any A,B,C∈ M .