A separably representable counterexample to Naimark's problem in ZFC

Abstract

We construct in ZFC a unital, simple, infinite-dimensional $C^*$-algebra whose nonzero irreducible representations form a single unitary equivalence class, but which admits a faithful representation on a separable Hilbert space. The algebra contains a unital copy of the canonical anticommutation relation (CAR) algebra, and the normalized CAR trace has a unique extension among all states. This extension is tracial, and its GNS representation is separable and faithful, with weak closure the hyperfinite $\mathrm{II}_1$ factor. In contrast, every nonzero irreducible representation acts on a Hilbert space of density $2^{\aleph_0}$. The construction separates the added unitaries into shell terms, controlled by finite CAR relations, and rank-one defect terms. These relations determine every irreducible representation of the generated algebra and every extension of the CAR trace. The algebra has norm density $2^{\aleph_0}$; consequently, the continuum hypothesis (CH) is equivalent over ZFC to the existence of a counterexample to Naimark's problem of norm density $\aleph_1$.