### - Art Gallery -

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 $${\displaystyle \diamondsuit }$$-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.
List of statements undecidable in $${\displaystyle {\mathsf {ZFC}}}$$