Compact Operators and Nest Representations of Limit Algebras

Abstract

In this paper we study the nest representations of a strongly maximal TAF algebra, whose ranges contain non-zero compact operators. We introduce a particular class of such representations, the essential nest representations, and we show that their kernels coincide with the completely meet irreducible ideals. From this we deduce that there exist enough contractive nest representations, with non-zero compact operators in their range, to separate the points. Using nest representation theory, we also give a coordinate-free description of the fundamental groupoid for strongly maximal TAF algebras. For an arbitrary nest representation $ρ$ of a strongly maximal TAF algebra, we show that the presence of non-zero compact operators in the range implies that the nest is similar to a completely atomic one. If, in addition, the range of $ρ$ is norm closed, then every compact operator in the range of $ρ$ can be approximated by sums of rank one operators in the range of $ρ$. In the case of $\bbN$-ordered nest representations, we show that the range of $ρ$ contains finite rank operators iff $\ker ρ$ fails to be a prime ideal.