Authors: Farah, Ilijas 
Hirshberg, Ilan
Vignati, Alessandro
Title: The Calkin algebra is ℵ1 -universal
Journal: Israel Journal of Mathematics
Volume: 237
First page: 287
Last page: 309
Issue Date: 1-Mar-2020
Rank: M22
ISSN: 0021-2172
DOI: 10.1007/s11856-020-2007-y
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.
Publisher: Springer Link

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