THE LATTICE OF IDEALS OF A TRIANGULAR AF ALGEBRA

We study triangular AF (TAF) algebras in terms of their lattices of cl osed two-sided ideals. Not (isometrically) isomorphic TAF algebras can have isomorphic lattices of ideals; indeed, there is an uncountable f amily of pairwise non-isomorphic algebras, all with isomorphic lattice s of ideals. In the positive direction, if A and R are strongly maxima l TAF algebras with isomorphic lattices of ideals, then there is a bij ective isometry between the subalgebras of A and R generated by their order preserving normalizers. This bijective isometry is the sum of an algebra isomorphism and an anti-isomorphism. Using this, we show that if the TAF algebras are generated by their order preserving normalize rs and are triangular sub-algebras of primitive C-algebras, then the lattices of ideals are isomorphic if and only if the algebras are eith er (isometrically) isomorphic or anti-isomorphic. Finally, we use comp lete distributivity to show that there are TAF algebras whose lattices of ideals can not arise from TAF algebras generated by their order pr eserving normalizers. Our techniques rely on constructing a topologica l binary relation based on the lattice of ideals, this relation is clo sely connected to the spectrum or fundamental relation (also a topolog ical binary relation) of the TAF algebra. (C) 1996 Academic Press. Inc .