Banach space theoretical construction of (primitive) spectra of $C^*$-algebras and the Naimark problem revisited

The Naimark problem asks whether $C^*$-algebras with singleton spectra are necessarily elementary. The separable case was solved affirmatively in 1953 by Rosenberg. In 2004, Akemann and Weaver gave a counterexample to the Naimark problem for non-separable $C^*$-algebras in the setting of ZFC $+~\diamondsuit_{\aleph_1}$, where $\diamondsuit_{\aleph_1}$ is Jensen's diamond principle. From this, at least, the affirmative answer to the Naimark problem can no longer be expected although a counterexample is not constructed in ZFC alone yet. In this paper, we study the difference between elementary $C^*$-algebras and those with singleton spectra, and find a property $P$ written in the language of closure operators such that a $C^*$-algebra is elementary if and only if it has the singleton spectrum and the property $P$. Banach space theoretical construction of (primitive) spectra of $C^*$-algebras plays important roles in the theory. Characterizations of type I or CCR or (sub)homogeneous $C^*$-algebras are also given. These results are applied to a geometric nonlinear classification problem for $C^*$-algebras.