A ZFC counterexample to Naimark's problem

What OpenAI claims

Gives an alternative to Tanaka's ZFC construction of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable. (Lean)

Paper

1. A counterexample to Naimark's problem in ZFC

   24 September 2026 · Preprint, not peer reviewed · PDF

   We construct in ZFC a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent. This gives a negative answer to Naimark's problem without additional set-theoretic assumptions.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.