Naimark's problem

Naimark's problem is a question in functional analysis asked by. 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). used the diamond 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 existence of a counterexample 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.