Researcher Of Preserving Problem And Derivable Mappings On Operator Algebras

The study of operator theory began in 30 times of the 20th century. With the fast development of the theory, it has become a hot branch playing the role of an initiator in modern mathematics now. It has unexpected relations and inter-infiltrations with quantum mechanics, noncommutative geometry, linear system and control theory, number theory as well as some other important branches of mathematics. In order to discuss the structure of operator algebras, many scholars both here and abroad have focused on mappings of operator algebras in recent years. For example, commutativity preserving maps, strong commutativity preserving maps, derivations, Lie derivations etc. They also have introduced some new concepts and new methods. For example, commuting mapping, functional identities. These mappings have become important tools in studying operator algebras at present time. The research of this thesis focuses on Lie mappings on factor von Neumann algebras, normal derivable mappings and orthogonal derivable mappings on B(H), preserving mappings and Jordan derivable mappings on factor von Neumann algebras. Using the technique of operator blocks, carrying on proper operator blocks to given operators according to content in research, inherent connection become more clear among operators through research them, more characters of discussed maps can be found. This article is divided into four chapters.In chapter I, the signification and background of this thesis selecting subject, are introduced. In addition, we offer necessary conceptions and conclusions for later chapters. Such conceptions and conclusions are with regard to derivations, bounded linear operators and the operator algebras.In chapter II, using the technique of operator blocks, we obtain the characteristics of nonlinear maps preserving Lie products on factor von Neumann algebras,we give the exact structure of nonlinear Lie derivations on factor von Neumann algebras, Lie-* derivations on factor von Neumann algebras, generalized*-Lie derivable mappings.In chapter III, we analyze the related characters of normal derivable mappings on B(H) at first. Then, unitary derivable mappings on B(H) are discussed and the structure of mentioned maps are obtained. the form of orthogonal derivable mappings on B(H) is obtained. At least, we consider Jordan orthogonal derivable mappings on B(H) In chapterⅣ,we give the exact structure of nonlinear strong commutativity preserv-ing-*maps on factor von Neumann algebras,additive maps preserving Schur-product. Furthermore,Jordan-* derivable mappings on factor von Neumann algebras are an-alyzed. Jordan derivable mappings at zero point orⅠpoint are discussed. At least, Jordan-* derivations on factor von Neumann algebras are analyzed.The results from the thesis consist of the following statements.(1)Let H be a complex separable Hilbert space with dim H≥2 and let M,N be two factor von Neumann algebras acting on H.Suppose thatΦ:M→N is a bijective map satisfyingΦ([A,B])=[Φ(A),Φ(B)]for all A,B∈M.Then there is a mapξ:M→CⅠwithξ(AB-BA):0 for all A,B∈M such that one of the following holds:(i)There exists an additive isomorphismψ:M→N such thatΦ(A)=ψ(A)+ξ(A) for all A∈M.(ii)There exists an additive anti-isomorphismψ:M→N such thatΦ(A)=-ψ(A)+ξ(A)for all A∈M.(2)Let M be a factor von Neumann algebra acting on a complex separable Hilbert space H with dim H>2.We prove that every nonlinear Lie derivation on a factor von Neumann algebra M is of the form A→φ(A)+h(A)Ⅰ,whereφ:M→M is an additive derivation and h:M→C is a nonlinear map with h[A,B]=0 for all A,B∈M.(3)Let M be a factor von Neumann algebra acting on a complex separable Hilbert space H with dim H>2.IfΦ:M→M be a Lie-* derivation,thenΦ(A)=AT-TA+ h(A)for all A∈M,where T∈M,h:M→CⅠis a linear mapping with h[A,B*]=0 for all A,B∈M and T+T*=βⅠ,β∈R.(4)LetΦ:B(H)→B(H)be a normal derivable linear map on B(H),thenΦ(A)= AT-TA+λA+f(A)Ⅰfor all A∈B(H),where T∈B(H),λ∈C,f:B(H)→CⅠis a linear mapping,and T+T*=βⅠ,β∈R.(5)Let H be a complex Hilbert space with dim H>2,B(H) denote the algebra of all linear bounded operators on H.LetΦ:B(H)→B(H)be a bounded linear mapping on B(H),ifΦ(A)*A+A*Φ(A)=Φ(A)A*+AΦ(A)*=Φ(Ⅰ)for any A∈B(H)with A*A=AA*=Ⅰ.thenΦ(A)=AS-SA for all A∈B(H),where S∈B(H),and S+S*=λⅠ,λ∈R.(6)Let H be a complex Hilbert space with dim H>2,B(H)denote the algebra of all linear bounded operators on H.We say that a linear mappingΦfromB(H)into B(H) is an orthogonal derivable linear mapping ifΦ(A)*B+A*Φ(B)=Φ(A)B*+AΦ(B)*=0 for any A,B∈B(H)with A*B=AB*=0.In this paper,we show the following result: every orthogonal derivable linear mapping on B(H)is a generalized inner derivation.(7)LetΦ:B(H)→B(H)be a Jordan orthogonal derivable bounded linear mapping on B(H),thenΦ(A)=AM-MA+λA for all A∈B(H),where M∈B(H),and M+M*=μⅠ,μ∈R,λ∈C.(8)Let M be a factor von Neumann algebra acting on a complex separable Hilbert space H with dim H>2.IfΦ:M→M be a Lie-* derivation,thenΦ(A)=AT-TA+h(A)for all A∈M,where T∈M,h:M→CⅠis a linear mapping with h(AB*一B*A)=0 for all A,B∈M and T+T*=βⅠ,β∈R.(9)Firstly, it is shown that linear surjective maps on matrix algebras which pre-serving Schur-product were a permutation operator. Using similar methods,preserving Schur-product additive bijection,preserving Schur-orthogonality linear maps in both di-rections,preserving Fan-product linear surjection maps on matrix algebras were studied respectively, we got similar results.It is deduced that Fan-product had similar property according to a series of properties of Schur-product.(10)LetΦ:Sa(H)→Sa(H)be a Jordan derivable linear mappings at zero point, thenΦ(A)=SA+AS*-λA for all A∈Sa(H),whereλ∈R and S∈B(H).Let M,N be two factor von Neumann algebras acting on H.Suppose thatΦ:M→N is a bounded Jordan derivable linear mapping at square zero point,thenΦis an inner derivation.