Some Derivable Mappings On Operator Algebras

The study of operator algebra theory began in 1930s. With the fast development of 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 quantum mechanics, noncommutative geometry, linear system and control theory, indeed number theory as well as some other important branches of mathematics. In order to discuss the structure of operator algebras, in recent years, many scholars both here and abroad have focused on linear mappings on operator algebras and have been introduced more and more new methods. For example, Jordan mappings, local mappings, 2-local mappings, dual local mappings, elementary mappings, linear preserving problems, generalized derivable mappings at zero point and so on were introduced successively, at present time these mappings have become important tool in studying operator algebras. In this paper, we mainly disscuss derivable mappings at zero point, Jordan derivations and 2-local derivations on some operator algebras. The details as following:In chapter 1, some notations, definitions are introduced and some theorems are given. In section I, we give some technologies and notations are introduced, such as the definitions of derivation, inner derivation, generalized derivation, generalized inner derivation, factor von Neumann algebra, subspace lattice, nest algebra and comparable element and so on. In section II, we give some given lemmas and some familiar propositions, theorems.In chapter 2, we first discuss the generalized derivable mappings at zero point on nest subalgebras of factor von Neumann algebras and on certain reflexive operator algebras whose lattices cointain a non-trivial comparable element. It is proved that every generalized derivable mapping at zero point on these algebras (no continuity is assumed) is a generalized derivation, respectively. They enrich Zhu's conclusion about the generalized derivable mappings at zero point on finite nest algebras from different methods. Subsequently, we discuss the generalized derivable mappings at zero point on standard operator algebras and show that every generalized derivable mapping at zero point on standard operator algebras is a generalized inner derivation.In chapter 3, we first present the notion of the Jordan derivable mappings at zero point and discuss the Jordan derivable mappings at zero point on nest subalgebras of factor von Neumann algebras and prove that every weakly continuous Jordan derivable mapping at zero point on nest subalgebras of factor von Neumann algebras is the sum of a derivation and a scalar multiplies identity mapping. More generally, we consider the Jordan derivable mappings at zero point on von Neumann algebras. Finally, we prove that every Jordan derivable mapping at zero point on B(H) is the sum of an inner derivation and a scalar multiplies identity mapping.In chapter 4, we discuss the Jordan derivations on certain reflexive operator algebras and the 2-local derivations on digraph algebras. In section I, we discuss the Jordan derivations on certain reflexive operator algebras Alg￡ whose lattices cointain a non-trivial comparable element and show that every Jordan derivation from Alg￡ into Alg￡—bimodule ￡ is a derivation. Thereby every linear 2-local derivation from Alg￡ into Alg￡—bimodule ￡ is a derivation. In section II, we discuss the 2-local derivations on digraph algebras, it is proved that every 2-local derivation from any symmetric digraph algebra into itself (no linearity is assumed) is a derivation. Moreover, we give an example to show that the conclusion may not be true if it has not the condition of symmetry.