Consistency of a counterexample to Naimark's problem

Details Consistency of a counterexample to Naimark's problem Consistency of a counterexample to Naimark's problem

2003-12-06

arxiv.org/abs/math/0312135v1

Mathematics

Operator Algebras

10 pages

Scientific paper

10.1073/pnas.0401489101

We construct a C*-algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses a combinatorial statement called the diamond principle, which is known to be consistent with but not provable from the standard axioms of set theory (assuming those axioms are consistent). We prove that the statement ``there exists a counterexample to Naimark's problem which is generated by $\aleph_1$ elements'' is undecidable in standard set theory.

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