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 ${\displaystyle *}$-representation up to unitary equivalence is isomorphic to the ${\displaystyle *}$-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 ${\displaystyle \diamondsuit }$-Principle to construct a C*-algebra with ${\displaystyle \aleph _{1}}$ generators that serves as a counterexample to Naimark's Problem. More precisely, they showed that the existence of a counterexample generated by ${\displaystyle \aleph _{1}}$ elements is independent of the axioms of Zermelo–Fraenkel set theory and the Axiom of Choice (${\displaystyle {\mathsf {ZFC}}}$).

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

See also

• List of statements undecidable in ${\displaystyle {\mathsf {ZFC}}}$ 
• 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, arXiv:math.OA/0312135, Bibcode:2004PNAS..101.7522A, doi:10.1073/pnas.0401489101, MR 2057719, PMC 419638, PMID 15131270
• 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