Naimark's problem

Naimark's problem is a question in functional analysis asked by {{harvs|txt|authorlink=Mark Naimark|last=Naimark|year=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). {{harvtxt|Akemann|Weaver|2004}} used the diamond principle to construct a C*-algebra with $\backslash aleph\_\{1\}$ generators that serves as a counterexample to Naimark's problem. More precisely, they showed that the existence of a counterexample generated by $\backslash aleph\_\{1\}$ elements is independent of the axioms of Zermelo–Fraenkel set theory and the axiom of choice ($\backslash mathsf\{ZFC\}$).

Whether Naimark's problem itself is independent of $\backslash mathsf\{ZFC\}$ remains unknown.