A separably representable counterexample to Naimark's problem in ZFC

@misc{tanaka2026separably,
  title = {{A separably representable counterexample to Naimark's problem in ZFC}},
  author = {Ryotaro Tanaka},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26930},
  eprint = {2609.26930},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26930},
}

TY  - UNPB
TI  - A separably representable counterexample to Naimark's problem in ZFC
AU  - Ryotaro Tanaka
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26930
UR  - https://arxiv.org/abs/2609.26930
AB  - 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 II_1 factor. In contrast, every nonzero irreducible representation acts on a Hilbert space of density 2^ℵ_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^ℵ_0; consequently, the continuum hypothesis (CH) is equivalent over ZFC to the existence of a counterexample to Naimark's problem of norm density ℵ_1.
ER  -