# Journal of Operator Theory

Volume 41, Issue 2, Spring 1999 pp. 223-259.

Classification of certain non-simple $C^*$-algebras**Authors**: Jakob Mortensen

**Author institution:**Department of Mathematics and Computer Science, Odense University, Campusvej 55, DK-5230 Odense M., Denmark

**Summary:**It is proved that the lattice of closed, two-sided ideals in a \cstaralgebra{} classifies the class of unital \cstaralgebra s which are inductive limits of sequences of finite direct sums of $\basicbb$ and have totally ordered lattice of ideals, up to $*$-isomorphism. Furthermore, it is proved that if the lattice of ideals of a separable, unital \cstaralgebra{} is totally ordered, then it is compact metrizable and has an isolated maximum in the order topology. Conversely, each totally ordered space (containing at least two points) which is compact metrizable and has an isolated maximum in the order topology appears as the lattice of ideals of a \cstaralgebra{} which is an inductive limit of a sequence of finite direct sums of $\basicbb$.

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