(α, β)-Multiplicative Derivation And Continuity Of Maps Between C~*-Algebras

This thesis mainly study the additivity of(α,β)-multiplicative derivations on rings and the continuity of *-preserving additive maps between C~*-algebra at the basis of Pierce decomposition theory.This thesis consists of two chapters.In the first chapter,We define(α,β)-multiplicative derivation and anti(α,β)-multiplicative derivation on rings,then we study the additivity of them and proved that(α,β)-multiplicative derivation and anti(α,β)-multiplicative derivation on rings,which own a nontrivial idempotent,are additive under some conditions In particular,We show that every(α,β)-multiplicative derivation on a prime ring with nontrivial idempotent is automatically additive.As applications,we show that each linear(α,β)-multiplicative derivation on M_n(C) is inner,and prove that each additive(α,β)-multiplicative derivation on M_n(C) can be expressed a sum of an(α,β)-inner derivation and an additive(α,β)-derivation induced by an additive derivation of C.Chapter two considering about the continuity of maps between C~*-algebras.We proved that the unital *-preserving additive map between C~*-algebras is contract if it 2-positive preserving.When unital *-preserving additive map also preserve positive,the map is contract just for self-adjoint elements.