Abstract

 The Cuntz Semigroup is an invariant for C-algebras combining K-theoretical and tracial information. It can be regarded as a C-analogue of the Murray-von-Neumann semigroup of projections of a von Neumann algebra. The Cuntz semigroup plays an increasingly important role in the classification of simple C-algebras. We propose a bivariant version of the Cuntz Semigroup based on equivalence classes of order zero maps between a given pair of C-algebras. The resulting bivariant theory behaves similarly to Kasparov's KK-theory: it contains the ordinary Cuntz Semigroup as a special case just as KK-theory contains K-theory and admits a composition product. It can be described in different pictures similarly to the classical Cuntz Semigroup and behaves well with respect to various stabilisations. Many properties of the ordinary Cuntz Semigroup have bivariant counterparts. Whilst in general hard to determine, the bivariant Cuntz Semigroup can be computed in some special cases. Moreover, it can be used to classify stably finite algebras in analogy to the Kirchberg-Phillips classification of simple purely infinite algebras via KK-theory. We also indicate how an equivariant version of the bivariant Cuntz Semigroup can be defined, at least for compact groups. If time permits, we discuss recent work in progress on a version of the Cuntz Semigroup for dynamical systems, more precisely, groups acting on compact spaces, with potential applications to classifiability of crossed products. (Joint work with Joan Bosa, Gabriele Tornetta.)