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By Glimm's dichotomy, a separable, simple $\textrm{C}^*$-algebra hascontinuum-many unitarily inequivalent irreducible representations if, and onlyif, it is non-type I while all of its irreducible representations are unitarilyequivalent if, and only if, it is type I. Naimark asked whether the latterequivalence 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 powerfuldiagonalization 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 necessarilyof type I. We show that this result is sharp by constructing an example of aseparably represented, simple $\textrm{C}^*$-algebra that has exactly twoinequivalent irreducible representations, and therefore does not satisfy theconclusion of Glimm's dichotomy. Our construction uses a weakening of Jensen's$\diamondsuit_{\aleph_1}$, denoted $\diamondsuit^\mathsf{Cohen}$, that holds inthe original Cohen's model for the negation of $\mathsf{CH}$. We also provethat $\diamondsuit^\mathsf{Cohen}$ suffices to give a negative answer toNaimark's problem. Our main technical tool is a forcing notion that genericallyadds an automorphism of a given $\textrm{C}^*$-algebra with a prescribed actionon its space of pure states.

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