Research On Derivable Mappings, Anti-derivable Mappings And Commuting Mappings

The study of operator algebra theory began in 1930s. With the fastdevelopment of the theory, now it has become a hot branch playing the role of aninitiator in morden mathematics. It has unexpected relations and interinfiltrationswith quantum mechanics, noncommutative geometry, linear system, control theory,and even number theory as well as some other important branches of mathematics.In order to discuss the structure of operator algebras, in recent years, a lot of schol-ars both here and abroad have focused on linear mappings of operator algebras andhave introduced more and more new methods. For example, local mappings, Jordanmappings, linear preserving problems, mappings derivable at zero point, commutingmappings, centralizing mappings and so on were introduced and researched succes-sively. At present time, these mappings have become important tools in studyingoperator algebras. In this paper, we mainly discuss derivable, anti-derivable mappings on Von Neumann algebra and commuting mappings in prime ring. The detailsas following:In chapterâ… , some notations, definitions are introduced and some well-knowntheorems are given. In section 1,we introduce some concepts, such as the definitionsof derivation, inner derivation, generalized derivation, generalized inner derivation,generalized Jordan derivation, Von Neumann algebra, prime ring and so on. Insection 2, we give some given lemmas that will be used in this paper.In chapterâ…¡, Firstly, we research the derivable mappings at unit and the anti-derivable mappings at zero point (unit) on Von Neumann algebra M. It is provedthat norm continuous linear mappings derivable at unit or anti-derivable at unit areinner derivations of M, that every norm continuous linear mapping anti-derivableat zero point is a generalized inner derivation of M. And we also prove that if Mis B(H), then every norm continuous linear mapping anti-derivable at zero point iszero, if M is B(H) and the dimention of H is infinity, then every norm continu-ous linear mapping anti-derivable at uint is zero. Secondly, we discuss generalizedderivable mappings at unit and Jordan derivable mappings at unit on Von Neumannalgebra, and we prove that every norm continuous linear mapping generalized deriv- able at unit is a generalized inner derivation, every norm continuous linear mappingJordan derivable at unit is a inner derivation.In chapterâ…¢, we discuss commuting mappings and centralizing mappings inprime ring. And we give a sufficient condition for a generalized derivation to be acommuting mapping in noncommutative prime ring whose characteristic is not tow.