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))=∑^nk=1pn(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→E(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A2A2=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))=∑^nk=1=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))=∑^nk=1 pn(A1,…,Ak-1,L(Ak),Ak+1,…,An)for all A1,A2,…,An∈M with A1 A2=0,then L(A)=φ(A)+f(A)for all A∈M,whereφ:M→M and f:M→E(M)(the center of M)are two mappings such that the restriction to PiMPj ofφis 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))=Σ^nk=1pn(A1,…,Ak-1,L(Ak),Ak+1,…,An)for all A1,A2,…,An∈with A1A2A1=0 if and only if L(A)=Φ(A)+h(A)I for all A∈M,whereΦis an additive derivation on M and is a functional of M such that h(pn(A1,A2,…,An))=0 for all A1,A2,…,An∈with A1A2A1=0.