Jordan Derivations Of Operator Algebras On The Map, And Jordan

(generalized) Jordan derivation and Jordan map are the important transformations of the operator algebras.They are also one of the wealthiest fields of operator algebras from 1950's.Many people have been studying the relationships between the(generalized) Jordan derivation and the(generalized) derivation;the additivity of the Jordan maps, because it is very important to reveal the structure of various operator algebras.In many instances,there are close connections between(generalized) Jordan derivation and(generalized) derivation;and the Jordan map remains additive.This connection has been investigated for some special algebras in recent years,and get a plentiful harvest.But so far,there is no characterization of the generalized Jordan derivation on upper triangular matrix algebras and triangular algebras,but also the additivity of Jordan maps in TUHF algebras.In this paper,we first solved that every generalized Jordan derivation from the algebra of all upper triangular matrices over a commutative ring with identity into its bimodule is the sum of a generalized derivation and an anti-derivation.Secondly,we prove that two forms of Generalized Jordan derivations are generalized derivations too.At last,we show that Jordan maps on THUF algebras are additive by making full use of the special structure of TUHF algebras and the "coordination".