Naimark's problem

Naimark's problem is a question in functional analysis. 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 Hilbert space.

The problem was solved in the affirmative for special cases (separable and type I C*-algebras). Akemann & Weaver (2004) used ◊ to construct a C^*-algebra serving as a counterexample to Naimark's problem and showed that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ₁ elements" is independent of the axioms of Zermelo-Fraenkel set theory and the axiom of choice (ZFC).

See also

• List of statements undecidable in ZFC
• Gelfand–Naimark theorem

External links

• 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, doi:10.1073/pnas.0401489101, MR2057719