Research On Some Mappings On Operator Algebras

The research on operator algebra originate in *-algebra which is comprised of bound-ed linearity operators on Hilbert space. It is mainly divided into two parts:to discuss the structure of operator algebra, and to discuss the classification problems. Since the struc-ture of operator algebra is very complicated, the classification problems of many operator algebras are not clear, such as von Neumann algebra and C*-algebra. So it is of great significance to study the effect of mapping of operator algebra on the structure. Preserver problem is a popular branch, which is a characterized problem of mappings which pre-serve a certain invariant. The purpose of research about preserver problems on operator algebra is to get some properties of the algebra by characterizing preserver mappings, and finally to realize the classification of operator algebra.In this thesis, we study mappings preserving Jordan multiple *-product, mappings p-reserving Jordan triple η-*-product, and weakly additive commuting mappings.By using methods of *-ring isomorphism and decomposition, we study the following three prob-lems:Firstly, the thesis characterizes the forms of bijections between two factor von Neu-mann algebras which are non-necessarily linear and preserve Jordan multiple *-product. Mappings which preserve Jordan *-product also preserve Jordan multiple *-product, but the reverse is not true in general, so the research on mappings preserving Jordan *-product is very significant. Let A and B be two factor von Neumann algebras.Let Φ be a non-necessarily linear bijection between A and B. It is shown that Φ preserves Jordan multiple *-product if and only if Φ is a *-ring isomorphism. In particular, if the von Neumann al-gebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism.Secondly, the thesis characterizes the forms of bijections between two von Neuman-n algebras(one of which has no central abelian projections)which are non-necessarily linear and preserve the Jordan triple η-*-product. Firstly, we define the Jordan triple η-*-product. Secondly, we study the additivity and linearly of mappings preserving Jordan triple η-*-product. Let η≠-1 be a non-zero complex number. Let Φ be a non-necessarily linear bijection between two von Neumann algebras (one of which has no central abelian projections)preserving the Jordan triple η-*-product with 0(I)=I. It is shown that if η is not real, then Φ is a linear *-isomorphism; if η is real, then φ is the sum of a linear *-isomorphism and a conjugate linear*-isomorphism.Finally, the thesis studies the forms of weakly additive commuting mappings from a algebra satisfying some assumptions to itself, and generalizes the result about additivity map.Let A be an algebra with unit 1 and an idempotent e.Let f be a weakly additive map from A to itself. It is shown that under some assumptions,if f is a commuting map, then for all x ∈A, there exist λ0{x)∈A and a map λ1 from A into Z(A),such that f(x)= Aq(x)x+λ1(x). As an application, a class of commuting weakly additive maps on Mn(F) are characterized.The results of this thesis reveal some important relations between mappings on the operator algebra and the properties of the operator algebra itself, which are helpful to understand the overall structure of the operator algebra and promote the development of differential equations, linear systems,quantum mechanics and so on.