2 8 A pr 2 00 5 COMPACT OPERATORS AND NEST REPRESENTATIONS OF LIMIT ALGEBRAS ELIAS

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