Characterizations Of The Local Properties Of Some Mappings On Operator Algebras

In this paper,we discuss the local properties of some mappings on operator algebras.The algebras that we study include von Neumann algebras,matrix algebras,triangular algebras,the algebra of operators on Hilbert C*-modules,and so on.The mappings that we consider include derivations,Lie derivations,centralizers,and so on.This paper splits into seven chapters.In Chapter one,we first introduce the background of this study.Then we give the problems that we would discuss and review the relevant developments and achievements until now.In the last,we give the definitions of the algebras and the mappings that we would mention in this paper.In Chapter two,we discuss 2-local derivations on matrix algebras and some special C*-algebras.Let A be a unital Banach algebra and M be a unital A-bimodule.We prove that if each Jordan derivation from A into M is an inner derivation,then every 2-local derivation from the algebra Mn(A)(n ? 3)into its bimodule Mn(M)is a derivation.And based on this,we give some corollaries,including that every 2-local derivation on a von Neumann algebra is a derivation.We also prove that every 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation.In Chapter three,we discuss local and 2-local Lie derivations on a kind of algebra satisfying some special conditions.On the algebras including factor von Neumann alge-bras,finite von Neumann algebras,nest algebras,UHF(uniformly hyperfinite)algebras,and the Jiang-Su algebra,we prove that every local Lie derivation is a Lie derivation.On the algebras including factor von Neumann algebras,UHF algebras,and the Jiang-Su algebra,we prove that every 2-local Lie derivation is a Lie derivation.Besides,for a finite von Neumann algebra which is not a factor,we construct an example of 2-local Lie derivation but not a Lie derivation.In Chapter four,we discuss the centralizable mappings on von Neumann algebras and triangular algebras.We prove that every point in a von Neumann algebra or a triangular algebra is a full-centralizable point.In Chapter five,we discuss the derivable mappings on von Neumann algebras.Let G be a given point in a von Neumann algebra A.We prove that every mapping which is derivable at G is a sum of a derivation and a centralizer.Moreover,G is a full-derivable point if and only if its central carrier is the unit.As a corollary,we prove that every nonzero element G in a von Neumann algebra A is a full-derivable point if and only if A is a factor von Neumann algebra.In Chapter six,we discuss the properties of the derivations on the algebra of operators on Hilbert C*-modules.Suppose that A is a commutative unital C*-algebra,and M is a full Hilbert A-module.We prove that every derivation on the algebra EndA(M)is A-linear,continuous,and an inner derivation.We also prove that every 2-local derivation on the algebras EndA(M)and EndA*(M)is a derivation.In addition,we characterize several A-linear mappings on EndA(M)satisfying some conditions for pairs(A,B)with AB = 0.In Chapter seven,we give some counter examples about the problems that we discuss in this paper,including nontrivial local and 2-local derivations,nontrivial local and 2-local Lie derivations,nontrivial centralizable mapping,and nontrivial derivable mapping.