Abstract: Meet irreducible ideals in direct limit algebras

Authors: Allan P. Donsig, Alan Hopenwasser, Timothy D. Hudson,
         Michael P. Lamoureux, Baurch Solel

We study the meet irreducible ideals in certain direct limit algebras.
The direct limit algebras will generally be strongly maximal triangular 
subalgebras of AF C*-algebras, or briefly, strongly maximal TAF algebras.
(All ideals are closed and two-sided.)

These ideals have a description in terms of the coordinates, or spectrum, 
that is a natural extension of one description of meet irreducible ideals 
in the upper triangular matrices. Additional information is available if the limit
algebra is an analytic  subalgebra of its \cstar-envelope or if the analytic
algebra is trivially  analytic with an injective 0-cocycle. In the latter case, we
obtain a complete description of the meet irreducible ideals, modeled on the
description in the algebra of upper triangular matrices. This applies, in
particular, to all full nest algebras.

An interesting subset of the meet irreducible ideals are the completely 
meet irreducible ideals, namely those satisfying an analogous condition, 
only for arbitrary intersections instead of just for finite intersections.
We describe these ideals and show that, for direct limit algebras 
generated by their order preserving normalizers, this subset is isomorphic
to the spectrum of the limit algebra (Theorem 5.3).
Also, there is a distance formula for ideals in a strongly maximal TAF
algebra (Theorem 6.2) that is analogous to Arveson's distance formula 
for nest algebras and to the distance formulae in \cite{MS2}.