Multiplicative Lie N-derivations On Von Neumann Algebras

Various derivations are one of important research topics in operator algebra and operator theory.In this paper,we mainly discuss multiplicative(generalized)Lie n-derivations on von Neumann algebras and give their characterizations from different ways.Let M be a von Neumann algebra without central summands of type I1.For any positive integer n,the(n-1)-th commutator of A1,A2,...,An ∈ is defined by Pn(A1,...,An)=[pn-1(A1,...,An-1),An],where p1(A1)=A1 and p2(A1,A2)=[A1,A2]is the usual Lie product.A map L:M→M called a multiplicative Lie n-derivation if L satisfies L(pn(A1,A2,…,An)=(?)(A1,…,Ak-1,L(Ak),Ak+1,…,An)(1)for all A1,A2,...,An ∈ a map G:M→M is called a multiplicative generalized Lie n-derivation if there is a Lie n-derivation L such that G(pn(A1,A2,···,An))=Pn(G(A1),A2,…,An)+(?)pn(A1,…,Ak-1,L(Ak),Ak+1,…,An).(2)k=2 holds A1,A2,···,An∈ Let P1∈ be a nonzero core-free projection,P2=I-P1.Assume that L:M→M and G:M→M are two maps and n>2 is any integer.It is shown that:1.if Eq.(1)holds for all A1,..An ∈ M with A1A2=0,then L(A)=φ(A)+f(A)holds for all A ∈ where φ:M→M is a map which is an additive derivation restricting to PiMPj,and f:M→Z(M)(the center of M)is a map satisfying f(pn(A1,A2,…,An))=0 for all A1,A2,…,An∈PiMPj with AIA2=0·Particularly,if M is a factor and n≥ 3,then Eq.(1)holds for all A1,...An ∈ M with A1A2A1=0 if and only if L(A)=φ(A)+h(A)I holds for all A ∈M,where:φ:M→M is an additive derivation and h:M→C is a functional such that h(pn(A1,A2,···,An))for all A1,A2,...,An ∈ M with A1A2A1=0;2.if there exists a map L satisfying Eq.(1)with A1A2=0 such that Eq.(2)holds for all A1,A2∈M with A1A2=0,then G(A)=τ(A)+h(A)holds for all A∈M,whereτ:M→M is a map which is an additive generalized derivation restricting to PiMPj(1 ≤i,j ≤2),and h:M^Z(M)is a map satisfying h(pn(A1,A2,···,An))=0 for all A1,A2,···,An ∈ M with A1A2=0.Particularly,G is a multiplicative generalized Lie n-derivation if and only if G(A)=φ(A)+f(A)holds for all A ∈M,whereφ:M→M is an additive generalized derivation and f:M→Z(M)is a central-valued map annihilating all(n-1)-th commutators.