-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
generators that serves as a counterexample to Naimark's problem.
More precisely, they showed that the existence of a counterexample generated by