Jordan Isomorphism And Elementary On Operator Algebras

The study of operator algebra theory began in 30s of the 20th century: With thefast development of the theory, now it has become a hot branch playing the role ofan initiator in modern mathematics. It has unexpected relations and inter infiltrationswith quantum mechanics, noncommutative geometry, linear system and control theory,number theory as well as some other important branches of mathematics. In order todiscuss the structure of operator algebra, in recent years, many scholars have focusedon linear maps and various multiplieative maps on opcrator algebras, and studied thealgebraic and geomrtrial properties and the problem of classification of these mapsMany deep and intercepting results have been achieved, and many new ideals and meanswas introduced. For example, Jordan multiplicative maps, Jordan triple multiplicativemaps, elementary maps, Jordan-triple elementary maps, Lie-skew multiplicative maps,local maps, 2-local maps, and so on, were. studied successively. At present time thesemaps have become important objectors and tools in studying operator algebras. Nestalgebras is a class of most importaut non-semisimple, non-prime and non-self adjointoperator algebras. Their finite dimensional models are upper triangular matrix algebras,but the infinite dimensional models are more complex. In this paper, we continue thestudy of Jordan multiplicative maps, Jordan triple multiplicative maps, elementarymaps, Jordan-triple elementary maps, Lie-skew multiplicative maps, Lie and Jordanderivation, 2-local isomorphism, local derivation and 2-local derivation on some operatoralgebrac. The main results are the following:1. We show that Jordan multiplicative bijective maps on prime operator algebrasmust be additive. As its application, we show that every Jordan * multiplicative bijec-tive map on every factor C~* algebra is a C~* isomorphism or a conjugate C~* isomorphism,in the special case of B(H), it must be a *-isomorphism or a conjugate *-isomorphism.We also prove the additivity of Jordan multiplicative bijective maps on nest algebras.2. We discuss the relation between elementary maps and ring isomorphisms, andwe give a characterization of elementary maps on stndard operator algebras on Banachspaces, JSL-algebras and nest algebras. For Jordan-triple elementeary maps, we provetheir additivity on a class of ring and show a relation of them with Jordan isomorphisms. Furthermore. we describe the Jordan elementary maps on standard operator algebrasand nest algebras. We also study the semi-Jordan elementary maps on effect algebrasand the space of self adjoint operators.3. We show that every Jordan triple multiplicative injective map on the space ofself-adjoint matrices must be surjective, and hence is a Jordan isomorphism. We alsostudy the Lie skew multiplicative bijective maps on B(H) (that is, the maps satisfyingΦ(AB- BA~*)=Φ(A)Φ(B)-Φ(B)Φ(A)~*), and show that these maps must be of theformΦ(A)=UAU~*, where U is a unitary or a conjugate unitary operator.4. We discuss the Lie, Lie-triple and Jordan derivations on nest algebras. Ourresults show that every Lie and Lie triple derivation must be of the formδ(A)+h(A)Iandτ(A) + g(A)I, where h,g satisfy respectively h[A, B]=0 and g[[A,B],C]=0,δandτare derivations. Every Jordan derivation is a derivation, and hence is an innerderivation.5. The 2-local isomorphism, local derivation and 2-local derivation on (?)-subspacelattice algebras (JSL algebras) are discussed. Our results show that 2-local isomor-phisms of JSL algebras must be isomorphisms; local derivations and 2-local derivationsof JSL algebras must be derivations.