The Cuntz semigroup, a Riesz type interpolation property, comparison and the ideal property

We define a Riesz type interpolation property for the Cuntz semigroup of a C*-algebra and prove it is satisfied by the Cuntz semigroup of every C*-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the C*-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite C*-algebras, and two of them are expressed in language of the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital C*-algebra that has (normalized) quasitraces. We prove that large classes of C*-algebras (including large classes of AH algebras) with the ideal property have this comparison property.