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
♢
-Principle to construct a C*-algebra with
ℵ
1
generators that serves as a counterexample to Naimark's Problem. More precisely, they showed that the statement "There exists a counterexample to Naimark's Problem that is generated by
ℵ
1
elements" is independent of the axioms of Zermelo-Fraenkel Set Theory and the Axiom of Choice (
Z
F
C
).
Whether Naimark's problem itself is independent of
Z
F
C
remains unknown.
