| Operator algebra is an important branch of modern mathematics. In recent years, In order to investigate the structure of operator algebras, many scholars have focused on the study of the linear mappings on them. Lots of deep and interesting results have been achieved, as well as a series of methods and skills. In this paper, we continue to study the linear mappings on operator algebras. The linear mappings we will discuss include derivations, Jordan derivations, centralizers, Lie derivations and the mappings associated with Hochschild 2-cocycles. The operator algebras we will discuss include Banach algebras and some non-adjoint reflexive algebras.We divide this paper into six chapters. In Chapter one, we recall some results about derivations and centralizers and introduce some preliminary concepts and necessary knowledge about operator algebras.In Chapter two, we study higher derivations and Jordan higher derivations. Let (?) be a commutative subspace lattice on H and A=alg(?). It is shown that every Jordan higher derivation from A into itself is a higher derivation. We also prove that if D=(δi)i∈N is a bounded higher derivable linear mapping at 0 from A into itself andδn(I)=0 for all n≥1, or D=(δi)i≤N is a higher derivable linear mapping at I from A into itself, then D=(δi)i∈N is a higher derivation. We also study the linear mappings D=(δi)i∈N satisfying on unital algebras and prove D=(δi)i∈N is a higher derivation under some conditions.Vukman defined a new type of Jordan centralizer named (m, n)-Jordan centralizer, motivated by this, in Chapter three, we study the mappings on unital algebras of the following form We consider the case of B=I or B=A. When B=I, we call this kind of mappings (m, n,l)-Jordan centralizer. When B=A, we call this kind of mappings (m, n,l)-Jordan triple centralizer. We will study these two kinds of mappings and prove that, under some conditions, (m, n,l)-Jordan centralizers and (m, n, l)-Jordan triple centralizer on semiprime rings or some reflexive algebras are centralizers.In Chapter four, we discuss generalized matrix algebras which contain triangular algebras. We will study the local actions of Lie derivations on U and prove that if L is a Lie derivable mapping at 0 or IA⊕0, then L is standard.In Chapter five, we study generalized Jordan derivations associated with Hochschild 2-cocycles which was first introduced by Nakajima. We will prove that if f is a gener-alized derivable mapping associated with Hochschild 2-cocycles at separating points on Banach algebras, then f is a generalized Jordan derivation associated with Hochschild 2-cocycles. We also show that if f is a generalized Jordan derivable mapping associ-ated with Hochschild 2-cocycles at 0 on CSL algebras, then f is a generalized derivation associated with Hochschild 2-cocycles.In Chapter six, we summarize the paper and pose some open questions. |
