Restrictions of pure states to subspaces of C⁎-algebras

Abstract

Given a unital subspace M of a $C^{⁎}$-algebra B, the fundamental problem that we consider is to describe those pure states ω on B for which $E_{\omega }=\left\{\omega \right\}$, where $E_{\omega }$ is the set of states on B extending $\omega {|}_M$. In other words, we aim to understand when $\omega {|}_M$ admits a unique extension to a state on B. We find that the obvious necessary condition that $\omega {|}_M$ 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 several variations on noncommutative peak points that have previously appeared in the literature. We illustrate that all of them are in fact distinct, address their existence and, in some cases, their relative abundance. To solve our main problem, we introduce a new type of peaking behaviour for ω, namely that the set $E_{\omega }$ be what we call an M-pinnacle set. Roughly speaking, our main result is that $\omega {|}_M$ admits a unique extension to B if and only if $E_{\omega }$ is an M-pinnacle set.