Let $\sK(H)$ be the $\C$-algebra of compact operators on a complex Hilbert space $H$. In 1948, Naimark proved that the representation theory of $\sK(H)$ is as easy as possible, and some years later, he asked whether this property characterizes $\sK(H)$ as a $\C$-algebra. Informally, a counterexample to Naimark's Problem is a $\C$-algebra whose representation theory is extremely easy but is not isomorphic to any algebra of compact operators.

It was not until 2004 when, assuming that $\diamondsuit$ holds, Akemann and Weaver exhibited the first counterexample to Naimark's Problem. All known counterexamples to Naimark's Problem has been constructed using some modification of the technique introduced by Akemann and Weaver, and it was not known whether $\diamondsuit$ was a necessary hypothesis to assure the existence of such a $\C$-algebra. In recent work with Ilijas Farah we give an answer to this question.