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Covering dimension of Cuntz semigroups II

    We show that the dimension of the Cuntz semigroup of a C-algebra is determined by the dimensions of the Cuntz semigroups of its separable sub-C-algebras. This allows us to remove separability assumptions from previous results on the dimension of Cuntz semigroups.

    To obtain these results, we introduce a notion of approximation for abstract Cuntz semigroups that is compatible with the approximation of a C-algebra by sub-C-algebras. We show that many properties for Cuntz semigroups are preserved by approximation and satisfy a Löwenheim–Skolem condition.

    Communicated by Yasuyuki Kawahigashi

    AMSC: Primary: 46L05, Primary: 46L85, Secondary: 54F45, Secondary: 55M10, Secondary: 46M20


    • 1. R. Antoine, F. Perera, L. Robert and H. Thiel, C-algebras of stable rank one and their Cuntz semigroups, Duke Math. J., in press, arXiv:1809.03984 [math.OA]. Google Scholar
    • 2. R. Antoine, F. Perera, L. Robert and H. Thiel , Edwards’ condition for quasitraces on C-algebras, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021) 525–547. Crossref, ISIGoogle Scholar
    • 3. R. Antoine, F. Perera and L. Santiago , Pullbacks, C(X)-algebras, and their Cuntz semigroup, J. Funct. Anal. 260 (2011) 2844–2880. Crossref, ISIGoogle Scholar
    • 4. R. Antoine, F. Perera and H. Thiel , Tensor products and regularity properties of Cuntz semigroups, Mem. Amer. Math. Soc. 251 (2018) viii+191. ISIGoogle Scholar
    • 5. R. Antoine, F. Perera and H. Thiel , Abstract bivariant Cuntz semigroups, Int. Math. Res. Not. IMRN (2020) 5342–5386. Crossref, ISIGoogle Scholar
    • 6. R. Antoine, F. Perera and H. Thiel , Abstract bivariant Cuntz semigroups II, Forum Math. 32 (2020) 45–62. Crossref, ISIGoogle Scholar
    • 7. R. Antoine, F. Perera and H. Thiel , Cuntz semigroups of ultraproduct C-algebras, J. Lond. Math. Soc. 102(2) (2020) 994–1029. CrossrefGoogle Scholar
    • 8. P. Ara, F. Perera and A. S. Toms , K-theory for operator algebras. Classification of C-algebras, in Aspects of Operator Algebras and Applications, Contemporary Mathematics, Vol. 534 (American Mathematical Society, 2011), pp. 1–71. CrossrefGoogle Scholar
    • 9. F. Borceux , Handbook of Categorical Algebra: Volume 1 Basic Category Theory, Encyclopedia of Mathematics and its Applications, Vol. 50 (Cambridge University Press, Cambridge, 1994). Google Scholar
    • 10. K. T. Coward, G. A. Elliott and C. Ivanescu , The Cuntz semigroup as an invariant for C-algebras, J. Reine Angew. Math. 623 (2008) 161–193. Crossref, ISIGoogle Scholar
    • 11. J. Cuntz , Dimension functions on simple C-algebras, Math. Ann. 233 (1978) 145–153. Crossref, ISIGoogle Scholar
    • 12. I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati and W. Winter , Model theory of C-algebras, Mem. Amer. Math. Soc. 271 (2021) viii+127. Google Scholar
    • 13. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott , Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications, Vol. 93 (Cambridge University Press, Cambridge, 2003). CrossrefGoogle Scholar
    • 14. L. Robert , Classification of inductive limits of 1-dimensional NCCW complexes, Adv. Math. 231 (2012) 2802–2836. Crossref, ISIGoogle Scholar
    • 15. L. Robert , The cone of functionals on the Cuntz semigroup, Math. Scand. 113 (2013) 161–186. CrossrefGoogle Scholar
    • 16. M. Rørdam and W. Winter , The Jiang–Su algebra revisited, J. Reine Angew. Math. 642 (2010) 129–155. Crossref, ISIGoogle Scholar
    • 17. H. Thiel , The topological dimension of type I C-algebras, in Springer Proc. Mathematics and Statistics Operator Algebra and Dynamics, Vol. 58 (Springer, Heidelberg, 2013), pp. 305–328. CrossrefGoogle Scholar
    • 18. H. Thiel, The generator rank of subhomogeneous C-algebras, preprint (2020), arXiv:2006.03624 [math.OA]. Google Scholar
    • 19. H. Thiel , Ranks of operators in simple C-algebras with stable rank one, Comm. Math. Phys. 377 (2020) 37–76. Crossref, ISIGoogle Scholar
    • 20. H. Thiel and E. Vilalta, Covering dimension of Cuntz semigroups, Adv. Math., in press, arXiv:2101.04522 [math.OA]. Google Scholar
    • 21. A. S. Toms , On the classification problem for nuclear C-algebras, Ann. Math. 167(2) (2008) 1029–1044. CrossrefGoogle Scholar