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.