Naimark's problem

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Naimark's Problem is a question in functional analysis. 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  \diamondsuit -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 statement "There exists a counterexample to Naimark's Problem that is 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.

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