Quasi Jordan Is Isomorphic To The Centralizer Zero ¦Ò-derivative Mapping

The study of operator algebra theory began in 30times of the 20th century.With the fast development of the theory,now it has become a hot branch playing the role of an initiator in modern 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 mappings on operator algebras and have been introduced more and more new methods.For example,local mappings,linear preserving problem, 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. On the basis of existing papers,in this paper we mainly discuss quasi-triple Jordan isomorphisms and quasi-triple Jordan*-isomorphisms of *-operator algebras,local centralizers andσ-derivable mappings at the point zero on nest algebras,and an identity on standard operator algebras.The article is divided into four parts and the details as following:In Chapter 1,some notations,definitions are introduced and some well-known theorems are given.In sectionⅡ,we introduce the definitions of quasi-Jordan isomorphism, quasi-triple Jordan isomorphism,left(right) centralizer,derivations,generalized derivation,σ-derivation and so on.In sectionⅢ,we give some well-known propositions.In Chapter 2,we mainly discuss quasi-triple Jordan isomorphisms and quasi-triple Jordan *-isomorphisms of *-operator algebras.Firstly,we prove that every quasi-Jordan isomorphism on algebra with involution is a Jordan isomorphism;Meanwhile,we also prove that every quasi-Jordan *-isomorphism is a Jordan *-isomorphism.Furthermore, we prove that every quasi-triple Jordan isomorphism on unital algebra with involution is an invertible element multiple of a Jordan isomorphism;at the same time,we also prove that every quasi-triple Jordan *-isomorphism is a unitary element multiple of a Jordan *-isomorphism.In Chapter 3,we pay our attention on linear mappings of nest algebras,we first prove that every linear local left(right) centralizer on nest algebra is a left(right) centralizer. Subsequently,we prove that every norm-continuous linearσ-derivable mappingδat the point zero on nest algebra is of the form:δ(A) =ψ(A)+λTA((?)A∈τ(N)),whereψbe aσ-derivation,T is a fixed invertible element ofτ-(N) andλis a fixed scalar.In Chapter 4,we discuss the additive mapφsatisfying 3φ(A~3) =φ(A)A~2+Aφ(A)A+ A~2φ(A) on standard operator algebras,we prove that every additive mapφsatisfying the above identity on standard operator algebra is of the form:φ(A)=λA,whereλis a fixed scalar.At the same time,we also prove that every additive map satisfying the same identity on semisimple H~*-algebra is a left and right centralizer.