Jordan Maps. Operator Algebra, Linear Mapping And Protection Of Leibniz's Rule

The study of operator algebra theory began in 30times years of 20 century. With the fast development of the theory, now it has become an important area in studying operator algebras. The study of maps on operator algebra is an important problem, it's important to understand and confer the structure of operator algebra, so people pay more attention to it.In recent years, with the deeply study of maps on operator algebra by many scholar home and abroad, for example, exchange map, Jordan map, elementary map and so on, now these maps have become strong tool for researching operator algebra. Jordan map is a kind of significant map on algebra or rings. Firstly in this paper we mainly and detailedly discuss the addition of Jordan maps on real symmetric matrices algebra and Jordan maps and Jordan-triple elementary map on Euclidean Jordan algebra, then it's to discuss the concrete form of linear map perserving Leibniz's rule on all-matrices algebra. The details as following:In chapter 1, we give some notions, definitions (for example, real symmetric matrices algebra, Euclidean Jordan algebra, Jordan map, Jordan-triple elementary map;all-matrices algebra, Leibniz's rule and so on)and some well-known knowledag and theorems.In chapter 2, we mainly discuss the addition of Jordan maps on real symmetric matrices algebra and Euclidean Jordan algebra. We prove that ifФ(aob)=Ф(a)oФ(b) for all a, b∈A or V,thenФis additive.In chapter 3, we discuss the concrete form of linear map perserving Leibniz's rule on all-matrices algebra. We prove that given:Mn(R)→Mn(R), if it's linear map perserving Leibniz's rule, then it mustbe exit A∈R, satisfying/j(A)= XA for any A∈Mn(R).