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

Abstract

We define a Riesz type interpolation property for the Cuntz semigroup of a $C^{\ast }$-algebra and prove it is satisfied by the Cuntz semigroup of every $C^{\ast }$-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the $C^{\ast }$-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite $C^{\ast }$-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 $C^{\ast }$-algebra that has (normalized) quasitraces. We prove that large classes of $C^{\ast }$-algebras (including large classes of $AH$ algebras) with the ideal property have this comparison property.

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