Ideals in Triangular Af Algebras

An ideal D is said to be join‐irreducible if whenever D = F ∨ G for ideals F and G, then either F = D or G = D. We study the class of join‐irreducible ideals in those strongly maximal triangular UHF algebras which arise as direct limits of triangular matrix algebras. Necessary and sufficient conditions are given for an ideal to be join‐irreducible in any such algebra. It is shown that semisimple triangular UHF algebras have no join‐irreducible ideals, whereas nest‐embedding algebras admit a large, tractable class of such ideals. The general case is more varied: in some algebras there are no join‐irreducible ideals but in others there are many; the key is shown to be an underlying property depending on the precise manner in which the inductive limit is constructed. Furthermore, we show that in many algebras, including the refinement algebra, no ideal of the form P U Q ⊥ for P and Q in Lat U is join‐irreducible. This is in contrast to the case of w * ‐closed ideals in nest algebras, where such ideals are the only ones which are w * ‐join‐irreducible.