An Operator Space Mapping Triple Jordan And Jordan Elementary Mapping

Preserver problems is always one of the most active research topic in operator algebra theory. Jordan maps, Jordan triple maps and Jordan elementary maps as a class of important preserver maps in operator algebra theory, they have been re-searched for years and some significant results have been showed to us. In this paper, we mainly discuss Jordan triple maps and Jordan elementary maps on symmetric operator spaces and self-adjoint operator spaces.This paper contains three chapters:In chapter 1, we give some notations, definitions, and some theorems which will be used in the later chapters. Firstly, we introduce some notations. Secondly, some definitions are given. They are conjugation, symmetric operator, Jordan triple algebra, Jordan triple map, Jordan elementary map and so on. Finally, we give some useful theorems.In chapter 2, we first dicuss a Jordan triple mapφon symmetric operator space. By proving the additivity of this map and using reference[l], we haveφ(A)=λXAXt,where Xt is transport of X.λ∈C\{0}, X:Hâ†'H is a bounded invertible linear or conjugate linear operator with Xt=X-1. Next, we prove the additivity of a Jordan triple mapφon self-adjoint operator space in the same way, and we also characterize the form of such map, which isφ(A)=βUAU*, whereβ∈R\{0}, U:Hâ†'H is a unitary or conjugate unitary operator.In chapter 3, we prove that if (M, M*) is a Jordan elementary map on two symmetric operator spaces, no matter whether these two symmetric operator spaces are different or not, so long as Mand M* are surjective, we can get that both M and M* are additive.