The Calkin algebra is ℵ1-universal

Ilijas Farah, Ilan Hirshberg, Alessandro Vignati

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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 20-universal C*-algebra. (ii) There exists a 20-universal C*-algebra, but Q(H) is not 20-universal, (iii) A 20-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.

Original languageEnglish
Pages (from-to)287-309
Number of pages23
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - 1 Mar 2020

ASJC Scopus subject areas

  • Mathematics (all)


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