The Calkin algebra is \(\aleph_1\)-universal (Q2190051)

Let \(\kappa\) be a cardinal. A \(C^*\)-algebra has density character \(\kappa\) if it has a dense subset of cardinality \(\kappa\) and \(\kappa\) is the minimal cardinal with that property. It is shown in this article that any \(C^*\)-algebra of density character \(\aleph_1\) embeds into the Calkin algebra. Here, \(\aleph_1\) is the smallest uncountable cardinal, and the Calkin algebra is the quotient of the bounded by the compact operators on a separable infinite-dimensional Hilbert space. A \(C^*\)-algebra of density character \(\kappa\) is \(\kappa\)-universal if any other \(C^*\)-algebra of density character \(\kappa\) embeds into it. In general, it is shown that there is no \(\kappa\)-universal \(C^*\)-algebra for cardinals \(\kappa<2^{\aleph_0}\). If the cntinuum hypothesis holds, that is, if \(\aleph_1 = 2^{\aleph_0}\), then the Calkin algebra is \(\aleph_1\)-universal. Without the continuum hypothesis, the following contradicting statements are all consistent with the ZFC axioms of set theory. First, that the Calkin algebra is \(2^{\aleph_0}\)-universal; secondly, that there is a \(2^{\aleph_0}\)-universal \(C^*\)-algebra but that the Calkin algebra is not \(2^{\aleph_0}\)-universal; and, thirdly, that there is no \(2^{\aleph_0}\)-universal \(C^*\)-algebra. In addition, it is shown that it is relatively consistent with the ZFC axioms that there is no \(\aleph_1\)-universal nuclear, simple \(C^*\)-algebra.