Given a unital $C^*$-subalgebra $A$ of a unital $C^*$-algebra $B$, a basic issue is to understand the extensions to $B$ of pure states on $A$. This underlies, for instance, the famous Kadison--Singer problem. In this talk, I will discuss the corresponding question when $A$ is replaced by an arbitrary (possibly non-selfadjoint) subalgebra. I will relate this to the existence of state excisions in the sense of Akemann--Anderson--Pedersen, as well as to non-commutative peaking behaviour.