Jordan (¦Á, ¦Â) - Derivations And Jordan-homomorphism

The study of operator algebra theory began in 1930s. With the fast developmentof the theory, now it has become a hot branch playing the role of an initiator in morden mathematics. It has unexpected relations and interinfiltrations with quantummechanics, noncommutative geometry, linear system, control theory, and even number theory as well as some other important branches of mathematics. In orderto discuss the structure of operator algebras, in recent years, a lot of scholars both here and abroad have focused on linear mappings of operator algebras and have introduced more and more new methods. For example, local mappings, derivations,Jordan derivations, generalized derivations, and generalized Jordan (α,β)-derivations and so on were introduced and researched successively. At present time, these mappings have become important tools in studying operator algebras. In this paper, firstly, we discuss Jordan (α,β)-derivations and generalized Jordan (α,β)-derivations on upper triangular matrix algebras, secondly, we prove the stability of Jordan homomorphisms and Jordan-derivations on unital Banach algebras, the stabilityof Jordan *- homomorphisms on unital C~*-algebras. The details as following:In chapterⅠ, we introduce some notations, definitions in this paper. Such as the definitions of derivations, Jordan derivations, generalized derivations, and generalized Jordan (α,β)-derivations and so on.In chapterⅡ, we prove that every generalized Jordan (α,β)- derivation from the algebra of all upper triangular matrices into its arbitrary bimodule is the sum of a generalized (α,β)-derivation and an (α,β)-antiderivation.In chapterⅢ, we discuss the stability of Jordan homomorphisms and generalizedJordan derivations on Banach algebras.In chapterⅣ, we discuss the stability of Jordan *-homomorphisms on C~*-algebras.