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?


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