According to the Elliott's Conjecture, a rather vast class of C*-algebras can be classified by means of K-theoretic data.
A counterexample due to Andrew Toms shows that the Cuntz Semigroup should be included in the set of Elliott's invariants.
KK-theory is another tool that can be employed for the purpouses of the classification of C*-algebras (cf. Kirchberg-Phillips).
The Bivariant Cuntz Semigroup is an attempt towards the definition of a new invariant, based on the idea of comparison of positive elements, as in the Cuntz semigroup, but with a bifunctor flavour, much like KK-theory. As completely positive maps of order zero induce maps between Cuntz semigroups, one can postulate that the right definition of such a new theory should be based on the introduction of an equivalence relation between such maps.
Under some simplifying assumption, one can carry out some standard simplification in the formulation of KK-theory. The open projection picture of the Cuntz Semigroup might offer a similar simplification for this new bivariant theory, so we have further explored its implications. From this analysis it emerged that the Scale of the Cuntz semigroup can be identified with the set of Cuntz-equivalence classes of positive elements from the positive cone of the C*-algebra.
Towards the Bivariant Cuntz Semigroup, Mathematisches Institut, Fachbereich Mathematik und Informatik der Westfälische Wilhelms-Universität Münster (January 14, 2013), Münster, Germany (slides).