Meet Irreducible Ideals and Representations of Limit Algebras

In this paper we give a criterion for an ideal of a TAF algebra to be meet irreducible. We show that an ideal $J$ of $A$ is meet irreducile if and only if the C${}^{\ast }$-envelope of the quotient $A\mathrm{/}J$ is primitive. In that case, A/J admits a nest representation which extends to a *-representation of the C${}^{\ast }$-envelope for $A\mathrm{/}J\mathrm{.}$ We also characterize the meet irreducible ideals as the kernels of nest representations; this settles the question of whether the n-primitive and meet irreducible ideals coincide.