Naimark's problem explained

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 (ZFC ).

Whether Naimark's problem itself is independent of

ZFC remains unknown.

See also

• List of statements undecidable in ZFC
• Gelfand–Naimark theorem