# Examples of pure state spaces of C*-algebras

The pure state space of $$C(X)$$ for compact Hausdorff $$X$$ is just $$X$$. The pure state space of $$M_n(\mathbb{C})$$ is given by the $$n$$-dimensional row vectors $$v$$ (normalised and up to a multiple) by $$\phi_v(a)=v A v^*$$. (This is Choi's theorem, and by using direct sums it also solves the case of finite dimensional $$C^*$$-algebras.) My question is: For what other examples of $$C^*$$-algebras is the pure state space actually known? What constructions can be used to help describe it or at least subsets of it?