2-local Derivations On Some Operator Algebras

This paper mainly studies the local property of some mappings on operator algebras,including 2-local derivations and approximately weak-2-local derivations on von Neumann algebras and C*-algebras,and so on.The article is divided into four chapters:The first chapter is the introduction,which mainly introduces the background of this study and review the relevant developments and achievements until now.Then we give the problems that we would discuss.The second chapter is preparatory knowledge,and gives some basic concepts.The third chapter is mainly about the approximately weak-2-local derivations on von Neumann algebras.We first define the approximately weak-2-local derivations and discuss some properties of the approximately weak-2-local derivations.In the last,it proves that every approximately weak-2-local derivation on a finite von Neumann algebra is a derivation.We generalize the result and prove that each approximately weak-2-local derivation on some C*-algebras is a derivation.The forth chapter is mainly about the 2-local derivation from a von Neumann algebra into its two-sided normal dual module.We first prove that each 2-local derivation from an abelian von Neumann algebra with separable predual and the B(H)that all bounded linear operators on Hilbert space H into its two-sided normal dual module is a derivation.In the last,if M the von Neumann algebra of type I has no infinite central summand,then every 2-local derivation from M into an arbitrary two-sided normal dual M-module is a derivation.