The Calkin algebra is $\aleph_1$universal
Abstract
We discuss the existence of (injectively) universal C*algebras and prove that all C*algebras of density character $\aleph_1$ 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 $2^{\aleph_0}$universal C*algebra. (ii) There exists a $2^{\aleph_0}$universal C*algebra, but $Q(H)$ is not $2^{\aleph_0}$universal. (iii) A $2^{\aleph_0}$universal C*algebra does not exist. We also prove that it is relatively consistent with ZFC that (iv) there is no $\aleph_1$universal nuclear C*algebra, and that (v) there is no $\aleph_1$universal simple nuclear C*algebra.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.01782
 Bibcode:
 2017arXiv170701782F
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Logic;
 46L05;
 03E35;
 03E75
 EPrint:
 18 pages. Some undefined LaTeX macros were removed from the abstract. This version is otherwise identical to v3. (The latter was a radically new version, with new coauthors, revised and updated)