In this chapter we present several results on the classification scheme of classical spaces, like \(C^\ast \) -algebras, spaces of operators and function spaces, using different techniques from the geometry of Banach spaces. Let \({\mathcal A}\) be a \(C^\ast \) -algebra, and for a Banach space X, let \({\mathcal L}(X)\) denote the space of bounded linear operators on X. The first part of the chapter is motivated by the works of Contreras et al., [6], Pfitzner [24] and [13]. We first study the geometry of the state space to classify the underlying space. We show that when \({\mathcal A}\) is a von Neumann algebra and the set of weak \(^\ast \) -continuous states is weakly compact, then \({\mathcal A}\) is finite dimensional. When \({\mathcal A}\) is a \(C^\ast \) -algebra, for each positive unit vector x, if the state space \(S_x=\{x^\ast \in {\mathcal A}^\ast : \|x^\ast \|=1=x^\ast (x)\}\) is weakly compact, then \({\mathcal A}\) is a CCR algebra. In the case of space of operators \({\mathcal L}(X)\) , we consider the question, when is \(S_I\) a weakly compact set? When the space of compact operators \({\mathcal K}(X)\) is an M-ideal in \({\mathcal L}(X)\) , we show that this happens only when X is a finite dimensional space. As an application, we study the relation between computing subdifferential limits of unit vector x in the norm to \(S_x\) being weakly compact. Our results give better estimates for subdifferential limits in the bidual of a space, and we give applications of these ideas in spaces of operators. Considering a Banach space X as canonically embedded in its bidual \(X^{\ast \ast }\) , we also study the stability of the structure of the state space of a unit vector \(x \in X\) when considered in \(X^{\ast \ast }\) . In the last section we use ideas from best approximation theory to classify CCR algebras as those \(C^\ast \) -algebras \({\mathcal A}\) for which in the dual, every weak \(^\ast \) -closed hyperplane is approximatively compact.