The Lattice of Ideals of a Triangular AF Algebra

We study triangular AF (TAF) algebras in terms of their lattices of closed two-sided ideals. Not (isometrically) isomorphic TAF algebras can have isomorphic lattices of ideals; indeed, there is an uncountable family of pairwise non-isomorphic algebras, all with isomorphic lattices of ideals. In the positive direction, if A and B are strongly maximal TAF algebras with isomorphic lattices of ideals, then there is a bijective isometry between the subalgebras of A and B 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 normalizers and are triangular subalgebras of primitive C*-algebras, then the lattices of ideals are isomorphic if and only if the algebras are either (isometrically) isomorphic or anti-isomorphic. Finally, we use complete distributivity to show that there are TAF algebras whose 0?lattices of ideals can not arise from TAF algebras generated by their order preserving 8?normalizers. Our techniques rely on constructing a topological binary relation based on the lattice of ideals; this relation is closely connected to the spectrum or fundamental relation (also a topological binary relation) of the TAF algebra.