Some Derivable Maps On Operator Algebras

The study of operator algebra theory began in 30times years of 20 century. Though compary with some other theory it is relatively new, but it has unexpected application in some mathematics theory and other subjects, such as quantum mechanics, noncommutative geometry, linear system, contral theory, number theory and some other important branches of mathematics. Accompany with its using in other subjects, this theory developed a lot. Now it has bocome a hot branch in mordan mathematics. The class of non-selfadjoint operator algebras is an important domain in operator algebras reaserching. And nest algebra is the most important kind in non-selfadjoint operator algebras. In recent years, many scholars both here and abroad have focused on them a lot. They have done many works, not only raising many new thinkings, but also introducing many advanced methods. In this paper we pay our attention on some maps on nest algebras and von Neumann algebras, such as the derivable maps and approximately derivable maps of nest algebras; Jordan semi-triple derivable maps of matrix algebras and B(H); the additivity ofγ-Jordan derivable maps of nest algebras. This paper contains four chapters. The main results as following:Chapter 1 mainly introduces some notations, definitions and some well-known theorems we will used in this paper. Firstly, we give some technologies and notations, and introduce the definition of nest algebra, von Neumann algebra, matrix algebra Subsequently we give some welt-known theorems.In chapter 2 we first discuss the additivity of derivable maps of nest algebras; We prove that every derivable maps of nest algebras is automatic additive. And then prove that every approximately derivable maps of nest algebras acting on an infinite dimensional Hilbert space is inner.In chapter 3 we first characterize the Jordan semi-triple derivable maps of matrix algebras. We prove that the Jordan semi-triple derivable maps of the algebra of all n×n matrices over a 2-torsion free commutative ring with unity is the sum of an inner derivation and A_φ, where A_φis the image of A underφapplied entrywise andφis an additive derivation over R. And then characterize the Jordan semi-triple derivable maps on the set of all linear bounded operators actingn on an infinite dimensional Hilbert space, then prove that it is additive and so is an inner derivation. In chapter 4 we discuss the additivity ofγ-Jordan derivable maps of nest algebras; We first prove that everyγ-Jordan derivable maps on B(H) is an additive derivation. And if H is an infinite dimensional Hilbert space, it is an inner derivation; Finally we get the same results when the nest is non-trivial.