The Calkin algebra is $\aleph_1$-universal

TL;DR

This paper proves that the Calkin algebra $Q(H)$ can embed all C*-algebras of density character $eth_1$, and discusses the relative consistency of the existence of universal C*-algebras of various sizes within ZFC.

Contribution

It establishes the $eth_1$-universality of the Calkin algebra and explores the consistency of the existence of universal C*-algebras under different set-theoretic assumptions.

Findings

$Q(H)$ embeds all C*-algebras of density $eth_1$.

The existence of $2^{eth_0}$-universal C*-algebras is independent of ZFC.

There is no $eth_1$-universal nuclear C*-algebra under certain assumptions.

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.