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A systematic approach for invariants of $C^*$-algebras

Volume 273 / 2023

Laurent Cantier Studia Mathematica 273 (2023), 63-99 MSC: Primary 46L35; Secondary 18B35. DOI: 10.4064/sm230516-22-6 Published online: 11 September 2023


We define a categorical framework in which we build a systematic construction that provides generic invariants for $C^*$-algebras. The benefit is significant as we show that any invariant arising this way automatically enjoys nice properties such as continuity, a metric on the set of morphisms and a theory of ideals and quotients which naturally encapsulates compatibility diagrams. Consequently, any of these invariants appear as good candidates for the classification of non-simple $C^*$-algebras. Further, most of the existing invariants could be rewritten via this method. As an application, we define a Hausdorffized version of the unitary Cuntz semigroup and explore its potential towards classification results. We indicate several open lines of research.


  • Laurent CantierDepartament de Matemàtiques
    Universitat Autònoma de Barcelona
    08193 Bellaterra, Spain
    Institute of Mathematics
    Czech Academy of Sciences
    115 67 Praha 1, Czechia

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