Naimark's problem

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 diamond principle to construct a C*-algebra with ${\mathrm{\aleph }}_1$ generators that serves as a counterexample to Naimark's problem. More precisely, they showed that the existence of a counterexample generated by ${\mathrm{\aleph }}_1$ 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