Consistency of a counterexample to Naimark's problem (with
C. A. Akemann).
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 N1 elements" is undecidable in
standard set theory.
PS, PDF, DVI, TeX files available here.
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