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) 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 ($\mathsf{Z}\mathsf{F}\mathsf{C}$).

Whether Naimark's problem itself is independent of $\mathsf{Z}\mathsf{F}\mathsf{C}$ remains unknown.

See also

• List of statements undecidable in $\mathsf{Z}\mathsf{F}\mathsf{C}$
• Gelfand–Naimark theorem

References

• Akemann, Charles; Weaver, Nik (2004), "Consistency of a counterexample to Naimark's problem", Proceedings of the National Academy of Sciences of the United States of America 101 (20): 7522–7525, doi:10.1073/pnas.0401489101, PMID 15131270, Bibcode: 2004PNAS..101.7522A 
• Naimark, M. A. (1948), "Rings with involutions", Uspekhi Mat. Nauk 3: 52–145 
• Naimark, M. A. (1951), "On a problem in the theory of rings with involution", Uspekhi Mat. Nauk 6: 160–164