Examples due to Villadsen, Rørdam, and Toms have shown that Elliott invariant is insufficient for the classification of all simple, separable, and nuclear C∗ -algebras. There are simple, separable, and nuclear C∗ -algebras that can be distinguished by their Cuntz semigroups but not by their Elliott invariant. In my talk, I will provide an overview of the Elliott classification program and introduce several related invariants, including the Elliott invariant and the Cuntz semigroup. I will explore the relationships between these invariants and show a characterization of the Cuntz semigroup and Cuntz comparison for a specific class of

non-simple C∗ -algebras with the ideal property. This characterization utilizes the lattice structure of the Murray-von Neumann semigroup and the cone of lower semicontinuous traces. This is a joint work with G. Elliott.