设ℳ和N是无I1或I2型中心直和项的von Neumann代数,其单位元分别为I和I′。本文证明非线性双射Φ:ℳ→N混合Lie可乘,即Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳ,当且仅当存在线性*-同构和共轭线性*-同构的直和Ψ:ℳ→N使得Φ(A)=Φ(I)Ψ(A),∀A∈ℳ,其中Φ(I)∈N是可逆中心元且Φ(I)2=I′。该结论将因子von Neumann代数上的非线性混合Lie可乘双射的结果推广到无I1或I2型中心直和项的von Neumann代数。Let ℳand Nbe von Neumann algebras with no central summands of type I1or I2, Iand I′be the identities of them. This paper proves that a bijective map Φ:ℳ→Nis mixed Lie multiplicative, that is, Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳif and only if Φ(A)=Φ(I)Ψ(A)for all A∈ℳ, where Ψ:ℳ→Nis a direct sum of a linear *-isomorphism and a conjugate linear *-isomorphism, Φ(I)is a central element in Nwith Φ(I)2=I′. The results about mixed Lie multiplicative maps on factor von Neumann algebras are generalized to von Neumann algebras with no central summands of type I1or I2.
Let M and N be two factor von Neumann algebras that their dimensions are large than 1,η≠-1 a non zero complex number and Φa(not necessary linear)bijection between two factor von Neumann algebras satisfying Φ(I)=I.For all A,B∈M,define by A■B=AB+BA the Jordan product of A and B,A·_(η)B=AB+ηBA^(*)the Jordan η-*-product of A and B,respectively.Let Φ and Φ^(-1)preserve the mixed Jordan triple η-*-products.It is proved that Φ is a linear *-isomorphism if η is not real and Φ is the sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η is real.
