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 diamond principle to construct a C*-algebra with generators that serves as a counterexample to Naimark's problem. More precisely, they showed that the existence of a counterexample generated by elements is independent of the axioms of Zermelo–Fraenkel set theory and the axiom of choice ().
Whether Naimark's problem itself is independent of remains unknown.