| Operator algebra is an important research field in functional analysis.Since F.J.Murray and J.von Neumann founded the theory of operator algebra in the 1930s,it has developed rapidly.Research on it not only has important theoretical value,but also has wide range of application prospects.It is closely related with differential geometry,number theory,linear systems,and quantum mechanics.Among them,von Neumann algebra is a very important class of self-adjoint operator algebras,which has a good development and a wide range of applications.Derivable maps on operator algebras are an important tool to study the construction of operator algebras.Thus research on it attracts attention of many scholars.This thesis is mainly studied four different types of nonlinear derivable maps on the factor von Neumann algebra.It is divided into five parts:In the first part,the development of operator algebra,the research status at home and abroad,and some basic knowledge are introduced.In the second part,the nonlinear*-Lie triple derivable map on the factor von Neumann algebra with the dimension greater than 1 in a complex Hilbert space is studied,and proves that it is a nonlinear*-Lie derivation.In the third part,the nonlinear mixed Jordan triple derivable map on the factor von Neumann algebra with the dimension greater than 1 in a complex Hilbert space is studied,and proves that it is an additive*-derivation.In the fourth part,the nonlinear bijection between two factor von Neumann algebras that preserves the mixed Jordan-product,where η≠-1 is studied.The characterization of this derivable map is given:there exists ε∈{-1,1},whenη∈R,εΦ is a linear or conjugate linear*-isomorphism;when η∈C\R,εΦis a linear*-isomorphism.In the fifth part,the nonlinear second-type mixed Jordan triple derivable map on the factor von Neumann algebra is studied.The characterization of this derivable map is given,and proves that it is an additive*-derivation. |
