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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