von Neumann代数上的局部可乘Lie n-导子
Local Multiplicative Lie n-derivations on von Neumann Algebras
A1,…,An的(n-1)-换位子记为pn(A1,…,An).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(pn(A1,…,An))=∑k=1n pn(A1,…,Ak-1,L(Ak),Ak+1,…,An)对所有满足条件A1A2=0的A1,A2,…,An ∈ 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PiMPj上是可加导子,f(pn(A1,A2,…,An))=0对所有满足A1A2=0的A1,A2,…,An ∈ PiMPj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(pn(A1,A2,…,An))=∑k=1n pn(A1,…,Ak-1,L(Ak),Ak+1,…,An)对所有满足A1A2A1=0的A1,A2,…,An ∈ 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(pn(A1,A2,…,An))=0对所有满足条件A1A2A1=0的A1,A2,…,An ∈ M成立.
Denote by pn(A1, …, An) the (n-1)-commutator of A1, …, An. 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(pn(A1, …, An))=∑k=1n pn(A1, …, Ak-1, L(Ak), Ak+1, …, An) for all A1, A2, …, An ∈ 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 PiMPj of φ is an additive derivation and f(pn(A1, A2, …, An))=0 for all A1, A2, …, An ∈ PiMPj 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(pn(A1, A2, …, An))=∑k=1n pn(A1, …, Ak-1, L(Ak), Ak+1, …, An) for all A1, A2, …, An ∈ 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(pn(A1, A2, …, An))=0 for all A1, A2, …, An ∈ M with A1A2A1=0.
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国家自然科学基金资助项目(11671006);山西省优秀青年基金资助项目(201701D211001)
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