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Abstract. Assuming the continuum hypothesis, we show that the Calkin algebra has 2 ℵ 1 outer automorphisms.

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 that these axioms are consistent). We prove that the statement ''there exists a counterexample to Naimark's problem which is generated by Ꭽ 1 elements'' is undecidable in standard set theory.