Centralizing Traces And Lie Triple Isomorphism Of Triangular Algebras

Triangular algebra are kind of important algebras. The importance depends on thenice good properties of themselves, but also that there are many important algebraswhich can be viewed as their special form, such as, upper triangular matrix algebras,nest algebras and so on.In this dissertation, we mainly study the centralizing traces of arbitrary bilinearmappings on triangular algebras and their applications in Lie theory and operatortheory. Especially, we focus on characterizing the form of Lie triple isomorphism oftriangular algebras. The explicit research target and conclusions are as follows.Firstly, we study the centralizing trace of bilinear mappings on triangular algebras.In this section, we use the methods of multiple linearization to resolve the problem.By elementary but not easy calculus, we can prove that the centralizing trace of anarbitrary bilinear mapping of triangular algebra is proper under the some mildconditions. This extends one theorem of Benkovic and Eremita concerningcommuting traces ([1, Theorem3.1]).Secondly, we study the Lie triple isomorphism of triangular algebras by theconclusion which is obtained in the first step. In this section, we consider the niceconnection between centralizing traces and Lie triple isomorphisms, which firstly isdiscovered by Bresar [2]. This connection makes it possible to describe theapproximate standard form of Lie triple isomorphisms of triangular algebras.Moreover, we can obtain that each Lie triple isomorphisms is of the standard form onupper triangular algebras and nest algebras.At the beginning of the90s Bresar described the form of commuting additive maps,and also the form of commuting traces of biadditive maps [2] on prime rings. Theseresults have initiated the theory of functional identities, which deals with maps ofrings satisfying some identical relations. There are very important theorem’s value indepicting the structures and properties of associative ring [3]. It is well known thatLie isomorphism and Jordan isomorphism are Lie triple isomorphisms. In this thesis,we provide the unified description of Lie isomorphism and Jordan isomorphism ontriangular algebras via the language of Lie triple isomorphisms. These result will become crucial part of the theory of functional identity of triangular algebras.