The Map Preserving The Lattices Of Invariant Subspaces And Centralizers On Operator Algebras

The study of operator algebra theory began in 30times of the 20th century. With the fast development of the theory, now it has become a hot branch playing the role of an initiator in morden mathematics. It has unexpected relations and interinfiltrations with quantum mechanics, noncommutative geomtry, linear sys-tem and control theory, number theory as well as some other important branches of mathematics. In order to discuss the structure of operator algebras, in recent years, many scholars both here and abroad have focused on mappings of operator algebras. For example, linear preserving mappings, centralizer, derivations etc,and they also have introduced some new concepts and new methods. For example, commuting mappings, functional identities etc.At present time these mappings have become important tools in studying operator algebras. In this paper we mainly discuss the additive map that preserve the lattices of invariant subspaces on upper triangu-lar matrix, some identities on finite rank operator algebras and standard operator algebras. This paper contains three chapters and the details as following:In chapter 1,some notations,definitions are introduced and some well-known results are given.In section 1.1,we introduce the definitions of upper triangular matrix algebra, standard operator algebras, self-adjoint operator algebras and finite rank operators.In section 1.2, we introduce the definitions of map preserving the lattices of invariant subspaces, prime ring and centralizer. In section 1.3,we give some well-known results.In chapter 2,we characterize the additive mapΦthat preserve the lattices of invariant subspaces on Tn.By describing the form of such maps,we haveΦ(A)=αA+<φ(A)I, where A∈Tn,α∈F andφ:Tn→F.In chapter 3,we first characterize a class centralizer of standard operator al-gebras A, we prove that if 2Φ(An+1)-Φ(A)An-AnΦ(A)∈FI or (s+t)φ(A3)-sφ(A2)A-tAφ(A2)∈FI holds for A∈A, thenΦis a centralizer; We also prove that if 2Φ(Am+n)=Φ(Am)An+AnΦ(Am) or (s+t)Φ(An+1)=sΦ(An)A+tAΦ(An) holds for A∈F(X),thenΦ(A)=λA, whereλis a fixed scalar.