Nsaf Algebra Irreducible Ideal

The meet irreducible ideals play an important role in the study of nonselfadjoint algebras.At first,we study the meet irreducible ideals of the finite direct sum of nest algebras,and give the forms of the meet irreducible ideals by the matrix units.We prove that in the finite direct sum of nest algebras the following are equivalent:(ⅰ)I is nest-primitive;(ⅱ)I is(K4);(ⅲ)I is meet-irreducible;(ⅳ)there exists a matrix unit e in A with/ = I(e)=(?){(?)is an ideal of A:e(?)}.Thus,we can give the setΩof meet irreducible ideals the hull-kernel topology,and for every(closed,two-sided) ideals I in the algebra,I is the intersection of all meet irreducible ideals containing I.We give the description of meet irreducible ideals in terms of matrix units(MI-chain) in the limit of a direct sum of nest algebras(NSAF algebras).For each MI-chain (e_k)k≥n,the ideal I associated with(e_k)is meet irreducible.Conversely,every proper meet irreducible ideal is induced by some MI-chain,but there is not a 1-1 correspondence between MI-chain and meet irreducible ideals.We also use the C~*-envelope of A/(?)to detect the meet irreducibility of the ideal(?).We show that(?)is meet irreducible if and only if the C~*-envelope of A/(?)is primitive.We also characterize the meet irreducible ideals as the kernels of bounded nest representation,i.e,the nest primitive ideal is meet irreducible.At last,we give the description of completely meet irreducible ideals in terms of CMI-chain,and characterize the completely meet irreducible ideals as kernels of nest representation of a special kind(CMI representation).