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 $*$ -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 $\diamondsuit$ -Principle to construct a C*-algebra with $\aleph _{1}$ 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 $\aleph _{1}$ elements" is independent of the axioms of Zermelo-Fraenkel Set Theory and the Axiom of Choice ( ${\mathsf {ZFC}}$ ).

Whether Naimark's problem itself is independent of ${\mathsf {ZFC}}$ remains unknown.

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, MR 2057719, arXiv:math.OA/0312135 , doi:10.1073/pnas.0401489101
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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Naimark's problem

See also

References