A Bivariant Theory for the Cuntz Semigroup

TL;DR

This paper develops a bivariant version of the Cuntz semigroup, extending its applicability and properties similar to KK-theory, and demonstrates its effectiveness in classifying certain C*-algebras.

Contribution

It introduces a new bivariant Cuntz semigroup, establishing its fundamental properties and applications in classification and invariants of C*-algebras.

Findings

Defined the bivariant Cuntz semigroup for various C*-algebras.

Proved properties like additivity, stability, and continuity.

Applied the theory to classify unital and stably finite C*-algebras.

Abstract

We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many properties formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C*-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we establish identities when tensoring with strongly self-absorbing C*-algebras. Finally, we show that the bivariant Cuntz semigroup of the present work can be used to classify all unital and stably finite C*-algebras.