Research On Several Classes Of Mappings On Operator Algebras

In this dissertation,we mainly study several classes of mappings on operator algebras.These mappings include bi-skew Lie derivations on factor von Neumann algebras,?-quasi-skew commuting mappings on von Neumann algebras,bilocal Lie derivations on nest algebras,Jordan ?g,h}-derivations on triangular algebras and skew Lie derivations on prime*-rings.This dissertation is divided into five chapters.In the first chapter,we introduce the significance and background of the selected topic,recall the present situation and achievements,and offer some necessary preliminary concepts and conclusions for later chapters.In Chapter 2,we investigate two classes of nonlinear mappings on von Neumann algebras.We obtain the characterization of nonlinear bi-skew Lie derivations on factor von Neumann algebras.Next,we give the concrete form of a class of nonglobal nonlinear?-quasi-skew commuting mapping on von Neumann algebras.The results are as follows.(1)Every nonlinear bi-skew Lie derivation on factor von Neumann algebras is an additive*-derivation.(2)Let A be a von Neumann algebra acting on a complex Hilbert space H and P(A)be the set of all projections in A.For any scalar? ?C\{0}.All nonlinear maps?:A?A satisfying[?(X),P]*?=[X,?(P)]*? for all X?A and all P?P(A)or P ?P(A)?{iI} are characterized.In Chapter 3,we investigate bilocal Lie derivations on nest algebras.We give the concrete form of bilocal Lie derivations on nontrivial nest algebras,and give the sufficient conditions for every bilocal Lie derivation to be a Lie derivation.Next,we prove that every bilocal Lie derivation on B(H)is a Lie derivation.The results are as follows.(3)Let N be a nontrivial nest on a complex separable Hilbert space H and AlgN be the associated nest algebra.Assume that ?:AlgN?AlgN is a bilocal Lie derivation.Then there exist an operator T?AlgN,a scalar ??C and a linear map f:AlgN? CI vanishing on each commutator such that ?(A)=[A,T]+?A+f(A)for all A?AlgN.(4)Let N be a non-atomic nest on a complex separable Hilbert space H and AlgN be the associated nest algebra.Then every bilocal Lie derivation from AlgN into itself is a Lie derivation.(5)Let N be a nontrivial nest on a complex separable Hilbert space H and AlgN be the associated nest algebra.If there exists an atom E of N with dim E>1,then every bilocal Lie derivation from AlgN into itself is a Lie derivation.(6)Let H be an arbitrary complex separable Hilbert space.Then every bilocal Lie derivation on B(H)is a Lie derivation.In Chapter 4,we investigate Jordan {g,h}-derivations on triangular algebras.We give the sufficient and necessary condition for every Jordan ?g,h}-derivation to be a {g,h}-derivation on triangular algebras.Next we give the characterization of every Jordan {g,h?-derivation to be a ?g,h}-derivation on nontrivial nest algebras.The results are as follows.(7)Let U be a triangular algebra.Then every Jordan ?g,h}-derivation f on U is a {g,h}-derivation if and only if g(1)? Z(U)or h(1)? Z(U).(8)Let N be a nontrivial nest on a complex separable Hilbert space H.Then every Jordan {g,h}-derivation on AlgN is a {g,h}-derivation if and only if dim0+?1 or dimH-??1.In Chapter 5,we investigate nonlinear skew Lie derivtions on prime*-rings,and prove that every nonlinear skew Lie derivation on unital 2-torsion free prime*-rings is an additive*-derivation.The results are as follows.(9)Let R be a unital 2-torsion free prime*-ring containing a nontrivial symmetric idempotent.We prove that if a map ?:R?R satisfies?([A,B]*)=[?(A),B]*+[A,?(B)]*for all A,B ?R,then ? is an additive*-derivation.