Compact operators and nest representations of limit algebras

In this paper we study the nest representations $\rho: \mathcal{A} \longrightarrow \operatorname{Alg} \mathcal{N}$ of a strongly maximal TAF algebra $\mathcal{A}$, 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 in $\mathcal{A}$. 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 $\rho: \mathcal{A} \longrightarrow \operatorname{Alg} \mathcal{N}$, we show that the presence of non-zero compact operators in the range of $\rho$ implies that $\mathcal{N}$ is similar to a completely atomic nest. If, in addition, $\rho (\mathcal{A} )$ is closed, then every compact operator in $\rho (\mathcal{A} )$ can be approximated by sums of rank one operators $\rho (\mathcal{A} )$. In the case of $\mathbb{N}$-ordered nest representations, we show that $\rho ( \mathcal{A})$ contains finite rank operators iff $\ker \rho$ fails to be a prime ideal.