Naimark's problem is a question in functional analysis asked by Naimark (1951). It asks whether every C*-algebra that has only one irreducible -representation up to unitary equivalence is isomorphic to the -algebra of compact operators on some (not necessarily separable) Hilbert space.
The problem has been solved in the affirmative for special cases (specifically for separable and Type-I C*-algebras). Akemann & Weaver (2004) used the -Principle to construct a C*-algebra with generators that serves as a counterexample to Naimark's Problem. More precisely, they showed that the statement "There exists a counterexample to Naimark's Problem that is generated by elements" is independent of the axioms of Zermelo-Fraenkel Set Theory and the Axiom of Choice ().
Whether Naimark's problem itself is independent of remains unknown.
- Akemann, Charles; Weaver, Nik (2004), "Consistency of a counterexample to Naimark's problem", Proceedings of the National Academy of Sciences of the United States of America, 101 (20): 7522–7525, arXiv: , doi:10.1073/pnas.0401489101, MR 2057719
- Naimark, M. A. (1948), "Rings with involutions", Uspehi Matem. Nauk, 3: 52–145
- Naimark, M. A. (1951), "On a problem in the theory of rings with involution", Uspehi Matem. Nauk, 6: 160–164
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