Researches On Some Certain Linear Mappings On Nest Subalgebras

The study of operator algebra theory begin in 20th century. Since it is used widely in mathematics and other scientific branches, it got great development at the beginning of the 20th century. Non-selfadjoint operator algebra is closely related to other mathematics branches, so it quickly becomes an important branche of operator algebras. Nest algebra is a class of most important non-semisimple and non-selfadjoint operator algebra. Its finite dimensional model is upper triangular matrix algebra, but the infinite dimensional model is more complex. In this paper, based on the conclusions of BreSar and Semrl, we discuss the linear mappings that are respectively Jordan derivation and generalized Jordan derivation actting on idempotents of nest algebras corresponding to arbitrary nests. At the same time, we discuss semi-local linear generalized derivations and linear kernel-range preserving mappings on nest subalgebras of factor von Neumann algebras and nest algebras, and we character generalized inner derivations of factor von Neumann algebra. Finally, we character the linear local mappings on a special matrix algebra. This paper contains four chapters.In chapter 1, we mainly introduce some notains and some well-known results. Firstly, we give some technologies and notations, and introduce the definitions of derivation, generalized derivation, local derivation, local generalized derivation, bilo-cal derivation, kernel-range preserving mapping, nest and nest algebra etc. Subsequently, we give some well-known results that are necessary and sufficient conditions of Jordan derivation and generalized Jordan derivation, respectively, and introduce some theorems on von Neumann algebra.In chapter 2, we discuss weakly continuous linear mappings from nest subal-gebra algMβ corresponding to arbitrary nest β of von Neumann algebra M into M and prove that if θ(E) = θ(E)E + Eθ{E) for all idempotent E ∈ algMβ, then θ is a Jordan derivation; and that if θ(E) = θ(E)E + Eθ(E) - Eθ(I)E, then θ is a generalized Jordan derivation . We get that the same conclusion holds for morm continuous linear mappings from nest algebra algβ corresponding to finite nest β into B(H). This generalizes the conclusions of Bresar and Semrl.In chapter 3, we analysis the linear semi-local generalized derivations and linear...