The Factor Von Neumann Algebra Preserves The Map Of The Mixed Lie Triple Product

The preserver problems about the map without additivity and linearity assum-ing which have attracted many author’s attentions.In this thesis,we define the first and second mixed Lie triple products,and consider the structure of the nonlinear map which preserving one of the products,by Peirce decomposition.The thesis is organized as follows:In Chapter 1,we fix some notation and recall necessary definitions needed in the thesis,such as factor von Neumann algebras,mixed Lie triple product and the second mixed Lie triple product and so on.In Chapter 2,we mainly discuss nonlinear maps which preserving the mixed Lie triple product on factors.Let M and N be factor von Neumann algebras on Hilbert space H with dim M,dim N>1.Let Φ:M→N be nonlinear map which satisfying Φ([[A,B]*,C])=[[Φ(A),Φ(B)]*,Φ(C)],for all A,B,C∈M.There exists ε∈{1,-1} such that Φ=εΨ.If dim M>4(or dim N>4),then Ψ is a linear or anti-linear*isomorphism.If dim M=dim N=4,then Ψ has one of forms:(1)Ψ(A)=UAU*;(2)Ψ(A)=UAU*;(3)Ψ(A)=-UAtU*+tr(A)I;(4)Ψ(A)=-UA*U*+tr(A)I,where U∈M2(C)is a unitary matrix.In Chapter 3,we mainly discuss nonlinear bijections which preserving the sec-ond mixed Lie triple product on factors.