The Studies Of Some Mappings On Triangular Algebras

The study of operator algebra theory began in 30times of the 20th cen-tury. With the fast development of the theory, now it has become a hot branch play-ing the role of an initiator in morden mathematics. It has unexpected relations andinterinfiltrations with quantum mechanics, noncommutative geomtry, linear systemand control theory, indeed number theory as well as some other important branchesof mathematics. In order to discuss the structure of operator algebras, in recentyears, many scholars both here and abroad have focused on mappings on operatoralgebras. They have introduced some new concepts and new methods. For exam-ple, modulo linear map, commuting map and functional identities etc. At presenttime these mappings have become important tools in studying operator algebras.Triangular algebra is a class of most important nonprime and nonselfadjoint op-erator algebra. Such as upper triangular matrix algebra and nest algebra are thisalgebra. On the basis of existing papers, in this paper we mainly and detailedly dis-cuss a class of nonlinear commuting maps(namely, modulo linear commuting maps),Jordan derivations and Generalized Jordan derivations on triangular algebras,σ-biderivations andσ-commuting maps, Generalized innerσ-biderivations and Gen-eralizedσ-commuting maps, Lie triple isomorphism on nest algebras. The detaialsas following:In chapter 1, some notations, definitions are introduced and some well-knowntheorems are given. In sectionⅡ, we introduce the definitions of triangular algebras,nest algebras, proper commuting map, Jordan derivations,σ-biderivation, Lie tripleisomorphism and so on. In sectionⅢ, we give some well-known theorems.In chapter 2, we disscuss nonlinear commuting maps (namely, modulo linearcommuting maps) on triangular algebras. By describing the form of such maps,we give a sufficient condition such that every modulo linear comnmting maps ontriangular algebras are proper. As an application, it is proved that every modulolinear commuting map on nest algebras is proper.In chapter 3, we first discuss Jordan derivations of triangular algebras andobtain that every Jordan derivation is derivation of triangular algebra, subsequently, we prove that every generalized Jordan derivation of triangular algebras is a sum ofa derivation and a generalized inner derivation.In chapter 4, we first discussσ-biderivations andσ-commuting maps on nestalgebras and we prove that everyσ-biderivation ofτ(N) is an innerσ-biderivationif dim 0_+≠1 or dim H_-~⊥≠1. As an application, we describe the form of mapf:τ(N)→τ(N) satisfying f(X)X=σ(X)f(X) for all X∈τ(N). subsequently,we discuss Generalized innerσ-biderivations and generalizedσ-commuting mapson nest algebras and we prove that every generalized innerσ-biderivation ofτ(N)is of the formφ(X,Y)=σ(X)AY+σ(Y)CX if dim 0_+≠1 and dim H_-~⊥≠1.As an application, we describe the form of maps f, g:τ(N)→τ(N) satisfyingf(X)X=σ(X)g(X) for all X∈τ(N).In chapter 5, we discuss Lie triple isomorphisms of nest algebras and we provethat every Lie triple isomorphism L:τ(N)→τ(M) is of the form L(x)=θ(x)+h(x), whereθis a isomorphism or the negative of an antiisomorphism and h is a mapfromτ(N)→CI such that h([[A,B], C])=0 for all A,B,C∈τ(N). At the sametime, we give an example that is not a Lie isomorphism but a Lie triple isomorphism.