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A ZFC counterexample to Naimark's problem
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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A counterexample to Naimark's problem in ZFC
Sep 24, 2026·naimark-counterexample-zfc.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.