By Glimm's dichotomy, a separable, simple 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 C*-algebras.In 2004, Akemann and Weaver gave a negative answer to Naimark's problem using Jensen's Diamond Principle diamond s1, a powerful diagonalization principle that implies the Continuum Hypothesis (CH). By a result of Rosenberg, a separably represented, simple 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 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 diamond s1, denoted diamond Cohen, that holds in the original Cohen's model for the negation of CH. We also prove that diamond 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 C*-algebra with a prescribed action on its space of pure states.& COPY; 2023 Elsevier Inc. All rights reserved.
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