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

TL;DR

This paper explores the properties of the Cuntz semigroup in $C^*$-algebras, introducing a Riesz interpolation property, characterizations of the ideal property, and a comparison notion for positive elements, with implications for various classes of $C^*$-algebras.

Contribution

It introduces a Riesz type interpolation property for the Cuntz semigroup and provides new characterizations of the ideal property in $C^*$-algebras.

Findings

The Cuntz semigroup of every $C^*$-algebra with the ideal property satisfies the Riesz interpolation property.

New characterizations of the ideal property are established in terms of the Cuntz semigroup.

Large classes of $C^*$-algebras with the ideal property have a comparison property for positive elements.

Abstract

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.