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The Calkin algebra is ℵ1-universal

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Abstract

We discuss the existence of (injectively) universal C*-algebras and prove that all C*-algebras of density character ℵ1 embed into the Calkin algebra, Q(H). Together with other results, this shows that each of the following assertions is relatively consistent with ZFC: (i) Q(H) is a 2ℵ0-universal C*-algebra. (ii) There exists a 2ℵ0-universal C*-algebra, but Q(H) is not 2ℵ0-universal, (iii) A 2ℵ0-universal C*-algebra does not exist. We also prove that it is relatively consistent with ZFC that (iv) there is no ℵ1-universal nuclear C*-algebra, and that (v) there is no ℵ1-universal simple nuclear C*-algebra.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, M3J 1P3, Canada

    Ilijas Farah

  2. Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva, 84105, Israel

    Ilan Hirshberg

  3. Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG) UP7D - Campus des Grands Moulins, Bâtiment Sophie Germain, 8 Place Aurélie Nemours, Paris, 75013, France

    Alessandro Vignati

Authors
  1. Ilijas Farah
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  2. Ilan Hirshberg
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  3. Alessandro Vignati
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Correspondence to Alessandro Vignati.

Additional information

I. H. and A. V.'s visit to Toronto were supported by NSERC. I. H. was supported by the Israel Science Foundation, grant no. 476/16. IF's visit to CRM was supported by the Clay Mathematics Institute. A. V. is supported by a PRESTIGE co-fund Scholarship and an FWO scholarship.

Part of this work was completed while the author was at the Katholieke Universiteit Leuven, Belgium.

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Farah, I., Hirshberg, I. & Vignati, A. The Calkin algebra is ℵ1-universal. Isr. J. Math. 237, 287–309 (2020). https://doi.org/10.1007/s11856-020-2007-y

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  • DOI: https://doi.org/10.1007/s11856-020-2007-y

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