IDEALS IN TRIANGULAR AF ALGEBRAS

An ideal S is said to be join-irreducible if whenever S = F v G for id eals F and G, then either F = S or G = S we study the class of join-ir reducible ideals in those strongly maximal triangular UHF algebras whi ch arise as direct limits of triangular matrix algebras. Necessary and sufficient conditions are given for an ideal to be join-irreducible i n any such algebra. It is shown that semisimple triangular UHF algebra s have no join-irreducible ideals, whereas nest-embedding algebras adm it a large, tractable class of such ideals. The general case is more v aried: in some algebras there are no join-irreducible ideals but in ot hers there are many; the key is shown to be an underlying property dep ending on the precise manner in which the inductive limit is construct ed. Furthermore, we show that in many algebras, including the refineme nt algebra, no ideal of the form PUQ(perpendicular to) for P and Q in Lat U is join-irreducible. This is in contrast to the case of w-close d ideals in nest algebras, where such ideals are the only ones which a re w-join-irreducible.