Local Characteristics Of Higher Derivations And Jordan Higher Derivations

The study of the relationship between derivations and Jordan Derivatives catches more and more attentions, and becomes one of the most attractive areas in operator algebra. K. R. Davidson made a literally introduction to the study of nest algebras in his book Nest Algebras, and proposed many innovative problems, which promoted the development of the next algebras, as well as that of nonlinear operators. During the recent year, many researchers from all over the world, discussed the relationships between the derivations and Jordan derivations on many kinds of operator algebras with various methods, and gained a series of results, which become an attractive area in the fields.For example, in 1957, Herstein proved the Jordan derivations mapping 2-torsion free prime rings onto itself is an derivations; In 2009, R. Alizadeh proved that every Jordan derivations mapping Mn(A) onto Mn(A) is a derivation; In 1990, D.R.Larson and A.R.Sourour studied the problems of local derivations and local isomorphism, and proved that every local derivation on B(X)is a inner derivation; in 1998, Zhang Jianhua proved that every Jordan derivation on nest algebras is a derivations; In 2007 Zhu Jun and Xiong Cangping proved every invertible element is all-derivative points in nest algebras; In 2008, Zhu Jun and Xiong Cangping proved that every non-zero element is an all-derivative point in upper triangular algebras; in the same year Zhu Jun and Xiong Cangping proved that every orthogonal projection operator is an all-derivable point on continuous nest algebras, where projection space belong to the nest.Researchers turned their efforts to the study of higher and generalized derivations when the theory of derivations and Jordan Derivations came into maturity.For instance, in 2011, Zeng Hongyan and Zhu Jun also proved the following results:(1) D=(δi)i∈N is a Jordan Derivations on Banach Algebras if Dis a higher derivative mapping at the invertible element X. (2) Every invertible element on nontrival nest algebras is higher an all-derivable points.The paper is made up of 5 chapters. In the first chapter, it is a brief introduction to the paper, including the notations, concepts, and basic properties and theorems. In the chapter 2, which is one primary part of the paper, we get the following results:Let A and B are rings with the identities I1 and I2, and M is (A, B)-bi-modulus, then T= (?):X∈AB,W∈M constructs a triangular algebras under common plus and multiply. We prove that the generalized Jordan higher derivations on triangular algebras are generalized higher derivations. In chapter 3, we obtain the result:the zeros are higher all derivatives points as dn(I)=0 in nest algebras. In chapter 4, we get: in von Neumann algebras, (1) the identity I is higher all-derivative points; (2) I is a Jordan higherall-derivative points; (3) the reversible elements Z are higher all-derivatives points. The chapter 5makes a conclusion to the paper and a prospect to the further study, and proposes problems to becontinued.