Naimark's problem
Naimark's problem is a question in functional analysis asked by Naimark (1951). It asks whether every C*-algebra that has only one irreducible [math]\displaystyle{ * }[/math]-representation up to unitary equivalence is isomorphic to the [math]\displaystyle{ * }[/math]-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) used the diamond principle to construct a C*-algebra with [math]\displaystyle{ \aleph_{1} }[/math] generators that serves as a counterexample to Naimark's problem. More precisely, they showed that the existence of a counterexample generated by [math]\displaystyle{ \aleph_{1} }[/math] elements is independent of the axioms of Zermelo–Fraenkel set theory and the axiom of choice ([math]\displaystyle{ \mathsf{ZFC} }[/math]).
Whether Naimark's problem itself is independent of [math]\displaystyle{ \mathsf{ZFC} }[/math] remains unknown.
See also
- List of statements undecidable in [math]\displaystyle{ \mathsf{ZFC} }[/math]
- Gelfand–Naimark theorem
References
- 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, doi:10.1073/pnas.0401489101, PMID 15131270, Bibcode: 2004PNAS..101.7522A
- Naimark, M. A. (1948), "Rings with involutions", Uspekhi Mat. Nauk 3: 52–145
- Naimark, M. A. (1951), "On a problem in the theory of rings with involution", Uspekhi Mat. Nauk 6: 160–164
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