The Calkin algebra is ℵ1-universal

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.