Lie Ideals Of Nest Subalgebras In Hyperfinite Factors

Many operator algebras scientists have been studying the Lie ideals of operator algebras since 1950's,and more people paid more attention to the operator algebras Lie structure,this is very important to reveal the structure of various operator algebras.In many algebras,the Lie ideals can be exactly determined;also there are close connections between the Lie ideal and the associative ideal of algebras. In this filed,we have got a plentiful harvest.In this paper,the description is first given for the faithful normal conditional expectation from a hyperfinite factor onto the diagonal of nest algebra of the hyperfinite factor.It is proved that center of the nest algebra is the scalar algebra.Second,It is decrypted the structure ofσ- weakly closed Lie ideals of regular nest subalgebras in hyperfinite factors,it is also revealed that the relation between the Lie ideal and the associative ideal.