von Neumann代数上的局部可乘Lie n-导子

A₁，…，A_n的（n-1）-换位子记为p_n（A₁，…，A_n）.令M是von Neumann代数，n ≥ 2是任意正整数，L：M → M是一个映射.本文证明了，若M不含I₁型中心直和项，且L满足L（p_n（A₁，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足条件A₁A₂=0的A₁，A₂，…，A_n ∈ M成立，则L（A）=φ（A）+f（A）对所有A ∈ M成立，其中φ：M → M和f：M → Z（M）（M的中心）是两个映射，且满足φ在P_iMP_j上是可加导子，f（p_n（A₁，A₂，…，A_n））=0对所有满足A₁A₂=0的A₁，A₂，…，A_n ∈ P_iMP_j成立（1 ≤ i，j ≤ 2），P₁ ∈ M是core-free投影，P₂=I-P₁；若M还是因子且n ≥ 3，则L满足条件L（p_n（A₁，A₂，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立当且仅当L（A）=φ（A）+h（A）I对所有A ∈ M成立，其中φ是M上的可加导子，h是M上的泛函且满足h（p_n（A₁，A₂，…，A_n））=0对所有满足条件A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立.

Local Multiplicative Lie n-derivations on von Neumann Algebras

Denote by p_n(A₁, …, A_n) the (n-1)-commutator of A₁, …, A_n. Assume that M is a von Neumann algebra, n ≥ 2 is any positive integer and L:M → M is a mapping. It is shown that, if M has no central summands of type I₁ and L satisfies L(p_n(A₁, …, A_n))=∑_k=1ⁿ p_n(A₁, …, A_k-1, L(A_k), A_k+1, …, A_n) for all A₁, A₂, …, A_n ∈ M with A₁A₂=0, then L(A)=φ(A) + f(A) for all A ∈ M, where φ:M → M and f:M → Z (M) (the center of M) are two mappings such that the restriction to P_iMP_j of φ is an additive derivation and f(p_n(A₁, A₂, …, A_n))=0 for all A₁, A₂, …, A_n ∈ P_iMP_j with A₁A₂=0 (1 ≤ i, j ≤ 2), P₁ ∈ M is a core-free projection and P₂=I -P₁; if M is a factor and n ≥ 3, then L satisfies L(p_n(A₁, A₂, …, A_n))=∑_k=1ⁿ p_n(A₁, …, A_k-1, L(A_k), A_k+1, …, A_n) for all A₁, A₂, …, A_n ∈ M with A₁A₂A₁=0 if and only if L(A)=φ(A) + h(A)I for all A ∈ M, where φ is an additive derivation on M and h is a functional of M such that h(p_n(A₁, A₂, …, A_n))=0 for all A₁, A₂, …, A_n ∈ M with A₁A₂A₁=0.