Nsaf Algebra And Can Not Be About Ideal

NSAF algebra is an important kind of the non-self-adjoint operator algebras which is introduced in the recent years.It contains the strongly maximal TAF algebra and the usual nest algebra.NSAF algebra is the inductive limit of the direct sum of the nest algebra(closed in the norm topology).The irreducible ideals(under the operations of join and meet) play an important role in the study of the non-self-adjoint algebras.This paper studies the join-irreducible ideals of the nest algebra.At first,we review the basic relations between the join-irreducible ideals and the meet-irreducible ideals,and then investigate the join-irreducible ideals of the standard and refinement algebras.Second,we study the join-irreducible ideals of the direct sum of the nest algebra;prove that the join-irreducible ideals of the direct sum of the nest algebras can be expressed by their own join-irreducible ideals.Further more,we study the join-irreducible ideals of the Nest UHF algebras,and prove that an ideal(?) in a Nest UHF algebra(?) is join-irreducible if and only if(?) is either an arbitrarily generated generalized semi-projection ideal or a semi-projection ideal which is principally generated by a partial isometry in a similar ideal space of some finite-dimensional factor.It lay over the result of Hudson.Finally,we give the definition of a JI-chain,establish the connection between the join-irreducible ideals and JI-chain,using the JI-chain characterize the join-irreducible ideals of the triangular algebra and the NSAF algebra.