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 (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.

See also

• List of statements undecidable in $\mathsf{ZFC}$
• Gelfand-Naimark Theorem