Restrictions of pure states to subspaces of $C^{\ast }$-algebras

Through the lens of noncommutative function theory, we study restrictions of pure states to unital subspaces of $C^{\ast }$-algebras, in the spirit of the Kadison–Singer question. More precisely, given a unital subspace $M$ of a $C^{\ast }$-algebra $B$, the fundamental problem is to describe those pure states $\omega$ on $B$ for which $E_{\omega }=\{\omega \}$, where $E_{\omega }$ is the set of states on $B$ extending $\omega {\mathrm{\mid }}_M$. In other words, we aim to understand when $\omega {\mathrm{\mid }}_M$ admits a unique extension to a state on $B$. We find that the obvious necessary condition that $\omega {\mathrm{\mid }}_M$ also be pure is sufficient in some naturally occurring examples, but not in general. Guided by classical results for spaces of continuous functions, we then turn to noncommutative peaking phenomena, and to the several variations on noncommutative peak points that have previously appeared in the literature. We perform a thorough analysis of these various notions, illustrating that all of them are in fact distinct, addressing their existence and, in some cases, their relative abundance. Notably, we find that none of the pre-existing notions provide a solution to our main problem. We are thus naturally led to introduce a new type of peaking behaviour for $\omega$, namely that the set $E_{\omega }$ be what we call a “pinnacle set”. Roughly speaking, our main result is that $\omega {\mathrm{\mid }}_M$ admits a unique extension to $B$ if and only if $E_{\omega }$ is an $M$-pinnacle set.