A bivariant theory for the Cuntz semigroup

Joan Bosa, Gabriele Tornetta, Joachim Zacharias

Research output: Contribution to journalArticleResearch


© 2019 The Authors 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 features 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 how to use the bivariant Cuntz semigroup of the present work to classify unital and stably finite C⁎-algebras.
Original languageEnglish
Pages (from-to)1061-1111
Number of pages51
JournalJournal of Functional Analysis
Issue number4
Publication statusPublished - 15 Aug 2019


  • Bivariant K-theory
  • Classification of C -algebras ⁎
  • Cuntz semigroup


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