∑math20

Operator algebras · Result 297

A ZFC counterexample to Naimark's problem

1 paperLean

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 1

First page of A counterexample to Naimark's problem in ZFC

A counterexample to Naimark's problem in ZFC

OpenAI · September 24, 2026

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.

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Cite
@misc{OAI:A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026,
  author = {{OpenAI}},
  title = {{A counterexample to Naimark's problem in ZFC}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf}{OAI:A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Naimark's problem asks whether a \(C^*\)-algebra whose nonzero irreducible representations are all unitarily equivalent must be an algebra of compact operators. The formalization gives a counterexample in ZFC: a unital infinite-dimensional simple complex \(C^*\)-algebra with a faithful tracial state has that uniqueness property for irreducible representations but is not isomorphic to the compact operators on any Hilbert space. No additional set-theoretic assumption is used.

Comparator links

ResultComparator statement
Counterexample to Naimark's problem in ZFCNaimark.lean

Scope note on GitHub →