von Neumann代数上的*-Lie三重导子

*-Lie Triple Derivation on von Neumann Algebras

摘要: 算子代数上的导子作为算子理论研究的一个重要部分，得到国内外学者的广泛关注。近年来，学者们相继引入Lie导子、*-Lie导子、*-Lie三重导子等映射。然而Lie导子是Lie三重导子，但反过来Lie三重导子不一定是Lie导子。本文研究和探讨von Neumann代数上的*-Lie三重导子是否为Lie导子。证明了在不含I₁型中心直和项的有限von Neumann代数之间的每一个非线性*-Lie三重导子是一个可加*-导子。

Abstract: As an important part of operator theory research, derivations on operator algebras have received widespread attention from scholars both domestically and internationally. In recent years, scholars have successively introduced mappings such as local Lie derivation, *-Lie derivation, *-Lie triple derivation. However, the Lie derivation is a Lie triple derivation; but conversely, the Lie triple derivation may not necessarily be a Lie derivation. This thesis studies and explores whether the *-Lie triple derivation on von Neumann algebra is a Lie derivation. We show that each nonlinear *-Lie triple derivation between von Neumann algebras without central summands of type I1 is an additive *-derivation.