Local (α,β)-Derivations On Operator Algebras

In this thesis, we study the relations of local (α,β)derivations and (α,β)de-rivations. The thesis consists of five sections.The first section is a brief introduction and the basic concepts and neces-sary preliminaries.In the second section, we show that every local (α,β) derivation from the matrix algebra Mn(C) into its Banach module is indeed an(α,β) derivation, whereα,βare linear mappings on Mn(C).In the third section, we show that every bounded local (α,β)derivation on an abelian von Neumann algebra (?) is an (α,β), whereαandβare bounded linear mappings on the abelian von Neumann algebra. Moreover, we also prove that whenαandβare multiplicative linear mappings such thatα(Ⅰ)=β(Ⅰ)=Ⅰ, every bounded local (α,β)derivation from an arbitrary von Neumann algebra into its Banach bimodule is an (α,β)derivation.In the fourth section, we show that, if (?) a unital standard operator algebra acting on a complex Banach space X, then every local (α,β) derivation from (?) into B(X) is an (α,β)derivation, whereαandβare linear multiplicative mappings such thatα(Ⅰ)=β(Ⅰ)=1.In the fifth section, we prove that,if A is a subalgebra of an AF C*-algebra E, which contains a canonical masa of E, then every bounded local (α,β) derivation from A into its module is an (α,β)- derivation, whereαandβare linear multiplicative mappings such thatα(Ⅰ)=1,β(Ⅰ)=Ⅰ.