Can you take Akemann--Weaver's $\diamondsuit_{\aleph_1}$ away?

Abstract

By Glimm's dichotomy, a separable, simple $\textrm{C}^*$-algebra has continuum-many unitarily inequivalent irreducible representations if, and only if, it is non-type I while all of its irreducible representations are unitarily equivalent if, and only if, it is type I. Naimark asked whether the latter equivalence holds for all $\textrm{C}^*$-algebras. In 2004, Akemann and Weaver gave a negative answer to Naimark's problem, using Jensen's diamond axiom $\diamondsuit_{\aleph_1}$, a powerful diagonalization principle that implies the Continuum Hypothesis ($\mathsf{CH}$). By a result of Rosenberg, a separably represented simple $\textrm{C}^*$-algebra with a unique irreducible representation is necessarily of type I. We show that this result is sharp by constructing an example of a separably represented, simple $\textrm{C}^*$-algebra that has exactly two inequivalent irreducible representations, and therefore does not satisfy the conclusion of Glimm's dichotomy. Our construction uses a weakening of Jensen's $\diamondsuit_{\aleph_1}$, denoted $\diamondsuit^\mathsf{Cohen}$, that holds in the original Cohen's model for the negation of $\mathsf{CH}$. We also prove that $\diamondsuit^\mathsf{Cohen}$ suffices to give a negative answer to Naimark's problem. Our main technical tool is a forcing notion that generically adds an automorphism of a given $\textrm{C}^*$-algebra with a prescribed action on its space of pure states.