Local Characterizations Of Derivation And Jordan Derivation

In recent years there has been considerable interest in studying which linear mapping on operator algebras are derivations, and we have obtained a lot of results. In 1990, the notion of local derivations was introduced independently by D. R. Larson and R. V. Kadison. Larson proved that every local derivation on B(H) is a derivation, where X is a Banach space, Kadison proved that every norm-continuous local derivation on a von Neumann algebra is a derivation. Zhang Jianhua showed that every Jordan derivable mapping in the triangular algebra is an inner derivation. Afterword, Wu Jing, Lu Shijie and Li Pengtong showed that every derivable mappingΦat 0 withΦ(I)=0 on nest algebras is an inner derivation; Zhu Jun and Xiong Changping proved that every norm-continuous generalized derivable mapping at 0 on finite CSL algebras is a generalized derivation, and every strongly operator topology continuous derivable mapping at unit operator I in nest algebras is an inner derivation (i.e. the unit operator I is an all-derivable point of nest algebras for the strongly operator topology). Lu Fangyan showed that in Banach space every idempotent is an all-derivable point.Lately, Zhu Jun and Xiong Changping have proved that (1) G∈TM2 is an all-derivable point of TM2 if and only if G≠0 where TM2 is the algebra of all n x n upper triangular matrices. (2) G∈Un is an all-derivable point in Un if and only if G≠0 where Un is the algebra of all n×n matrices. In 2008, Jing Wu has proved that I is a Jordan all-derivable point ofB(H). In 2007, An Runling and Hou Jinchuan have proved that some idempotent are all-derivable points on rings, next year they have proved these idempotent are Jordan all-derivation points. Enlightened and guided by these results, we extend the results which Zhu Jun and Xiong Changping have proved that every point is a Jordan all-derivation point.It has three chapters in this article. The first chapter introduces notions, notes and properties. The second chapter is the main body that discusses the problem of Jordan full-derivation and full derivable points. The paper has proved: is a all derivable point; and is a Jordan all-derivable point。In section 3, Based on the theorem that G is a full derivable point of the upper train gular algebra if and only if G≠0 by Zhu Jun and the following definition of a Jorda n derivable mapping,we get some Results of Jordan all-derivale in the upper algebra. For (?)S,T(?)H withST=P,we haveΦ(ST+TS)=Φ(S)T+SΦ(T)+TΦ(S)+Φ(T)S, then calledΦ:A→A is Jordan derivable at P.