Operator Lie Algebra

The theory of operator algebras' Lie structure is one of the most wealthiest fields of operator algebras from 1950's. Many people have been studying the Lie structure(Lie ideals, Lie derivations, Lie isomorphism) because it is very important to reveal the structure of various operator algebras.In many instances, the Lie ideals can be exactly determined, or there are close connections between the Lie ideal structure and the associative ideal structure of algebras. This connection has been investigated for some special algebras in recent years, and get a plentiful harvest. In the case of non-self-adjoint operator algebras, the weakly closed Lie ideals in Nest algebras, the norm-closed Lie ideals in TUHF algebre and TAF algebras have been fulfilled. Marcoux has determined that there are only four closed ideals in UHF algebras. But so far, there is no characterization of the closed Lie ideals in AF C*-algebras. In this paper, we first solved the characterization of the Lie ideals in GICAR algebra, which is an important AF C*-algebra. We will find GICAR algebra, different from UHF algebra, has rich closed Lie ideals.AF C*-algebra is the self-adjoint limit algebra, so it has relatively complex structure. Thus there are some difficulties for the characterization of the Lie ideals in general AF C*-algebras. In the second chapter of this paper, we study the closed Lie ideals in AF C*-algebras recur to the Groupiod theory.At last, this paper gives a description of the relation between Lie triple derivation and associative derivations of TUHF algebras. By the defination of Lie triple derivation, naturally, it is in close connection with the associative derivation. C.R.Miers has shown that the Lie triple derivation on a vN algebra M with no central abelian summands has the form D + λ, where D is an associative derivation and A is a linear map of M into its center which annihilates brackets of operators. Making full use of the special structure of TUHF algebras and the method of " coordinization " , we show that the continuous Lie triple derivation on TUHF algebra has the analogous result. Consequently, the Lie derivation on TUHF algebra also has this form.