Additive Rank Preserving On Matrix Algebra

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 geometry, linear system and control theory, indeed 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 linear mappings on operator algebras and have been introduced more and more new methods. For example, elementary mappings, linear preserving problems and so on were introduced successively, at present time these mappings have become important tool in studying operator algebras. But the study of the additive preserver problems are very important in operator or matrix theory. To solve these preserving problems, it is a common path to reduce the given questions to questions of characterizing additive maps which preserve ranks, or decrease ranks, or preserve rank-one nipotents, or preserve rank-one idempotents. These questions have been discussed by many mathematicians on Banach space cases, and many deep results were obtained. In this paper we mainly and detailedly discuss additive rank one preserving surjections on Hermitian matrix spaces H_n(C), the counter-tripotent preserving linear operators from alternate matrix spaces K_2n(F) to all matrix M_m($) over a field F with chF ≠ 2,3 and 2n ≤ m, and additive the least rank preserving from symmetric matrix spaces to alternate matrix spaces over a field. The details as following:In chapter 1, let H_n(C) be the Hermitian matrix spaces on C, We characterize additive rank-one preserving surjections on Hermitian matrix spaces H_n(C), obtain the form of φ when φ is an additive rank-one preserving surjections on Hermitian matrix spaces H_n(C), give the form of φ when φ preserves the invertibility of H_n^*(C), and the form of φ when φ preserves the determinant.In chapter 2, suppose F is a field of characteristic not 2,3, we discuss the counter-tripotent preserving linear operators from alternate matrix spaces K_2n(F) to all matrix spaces M_m(F) over a field F and 2n ≤ m, give the form of T when T is the counter-tripotent preserving linear operators from alternate matrix spacesto all matrix spaces Afm(F), and the counter-tripotent preserving linear operators from alternate matrix spaces /sT2n(F) to alternate matrix spaces Km($) over a field F with chF ^ 2,3 and 2n < ra are also characterized.In chapter 3, suppose F is a field of characteristic not 2, let Sn($) be symmetric matrix spaces on F , we discuss the additive the least rank preserving from symmetric matrix spaces to alternate matrix spaces over a field F, and the form of $ that the additive the least rank preserving from symmetric matrix spaces to alternate matrix spaces over F is given.