Nonlinear Jordan Higher Derivations Of Triangular Algebras

Let A be an associative algebra. For any x,y∈A, we define two operations by [x,y]=xy-yx and xoy=xy+yx. Then (A,[,]) is in fact a Lie algebra and (A,o) is a Jordan algebra. To study the relationship and classification of the associative structure, Lie structure and Jordan structure of A is of crucial importance. We often use the linear mappings (or generally speaking, operator theory) to study relevant algebras. In particular, we mainly study the Jordan structure of triangular algebras in this dissertation. Triangular algebras are a kind of important algebras. The importance depends on the nice properties of themselves, but also that there are many important algebras which can be viewed as special form of triangular algebras, such as, upper triangular matrix algebras、block upper triangular matrix algebras、nest algebras and so on.We first introduce the definitions and some basic properties of triangular algebras and (nonlinear) Jordan derivations. Then some typical examples of triangular algebras are listed out. Furthermore, we get the conclusion that any nonlinear Jordan derivation on a2-torsion free triangular algebra is in fact an additive derivation. Then we discuss the nonlinear Jordan higher derivations of triangular algebras, and extend the main results of nonlinear Jordan derivations to the general higher order case, i.e. each nonlinear Jordan higher derivation of a2-torsion free triangular algebra degenerates to an additive higher derivation.Nest algebras, coming from the operator theory of the field of analysis, are non-self-adjoint and non-semiprime. It is well-known that, trivial nest algebras are von Neumann prime algebras, and non-trivial nest algebras belong to the so called triangular algebras. The most interesting fact of this thesis is the generalization of a theorem (Christensen Theorem [1]) in functional analysis to the nonlinear case by pure algebraic computation. More precisely, we proved that every nonlinear Jordan derivation of nest algebras over an infinite dimensional Hilbert space is an inner derivation.