Meet-irreducible ideals and representations of limit algebras

In this paper we give criteria for an ideal J of a TAF algebra A to be meet-irreducible. We show that J is meet-irreducible if and only if the C∗-envelope of A/J is primitive. In that case, A/J admits a faithful nest representation which extends to a ∗-representation of the C∗-envelope for A/J. We also characterize the meet-irreducible ideals as the kernels of bounded nest representations; this settles the question of whether the n-primitive and meet-irreducible ideals coincide.