Operator Algebras Of Lie Ideals

The theory of operator algebras' Lie structure is one of the wealthiest fields of operator algebras from 1950's. Many people have been studying the Lie structure (Lie ideal, 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 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-ad joint operator algebras, the weakly closed Lie ideals in Nest algebras, the norm-closed Lie ideals in TUHF algebras and TAF algebras have been fulfilled. Marcoux has determined that there are only four closed Lie ideals in UHF algebras. But so far, there is no characterization of the closed Lie ideals in AF vN algebras and Groupoid C~* -algebra.In this paper, we first solved the characterization of Lie ideal in aσ-weakly closed triangular subalgebra A of an AF vN algebra B with cartan subalgebra D. Second, we give a detailed description of the structure of closed Lie ideals in a finite CSL algebra Alg(M) of simple Groupoid C~*-algebra B.