Generalized Anti-differentiable Mapping And Linear Mapping

The study of operator algebra theory began in 30's of the 20th century. With the fastdevelopment of the theory, now it has become a hot branch playing the role of an initiatorin modern mathematics. It has unexpected relations and interinfiltrations with quantummechanics, noncommutative geometry, linear system and control theory, indeed numbertheory as well as some other important branches of mathematics. In order to discuss thestructure of operator algebras, in recent years, many scholars both here and abroad havefocused on mappings on operator algebras. They have introduced some new conceptsand new methods. For example, modulo linear map, commuting map; Lie map; linearpreserving map; map derivable at zero(unit) point and functional identities etc. At presenttime these mappings have become important tools in studying operator algebras. In thispaper we mainly discuss linear maps satisfying identity on standard operator algebras;generalized anti-derivable maps on von Neumann algebras and linear maps preservingJ-idempotents on upper triangular matrix algebras. The details as following:In Chapter 1, we mainly discuss the linear mapΦssatisfy identitΦy(A～4)=Φ(A)A～3 +AΦ(A)A～2 + A～2Φ(A)A + A～3Φ(A) on standard operator algebras, we prove that exist T GB(H) such thatΦis of the form Aâ†'AT-TA and generalize this result.In Chapter 2, we first research the norm continuous linear mapping generalized anti-derivable at (unit)zero point on von Neumann algebras and prove it is generalized innerderivation on von Neumann algebras, subsequently, we prove that norm continuous linearmapping generalized Jordan derivable at unit point is generalized inner derivation of vonNeumann algebras.In Chapter 3, we give the concrete form of linear bijective maps preserving/-idempotents onupper triangular matrix algebras, that is one of the following four forms:(1)(?)(2)(?)(3)(?)(4)(?)where (?) is a linear function (?) is an invertible matrix of (?),J denotethe n×n matrix J=(?) where (?), is the Kronecker delta function.