  We study boundaries for unital operator algebras. These are sets of
irreducible $*$-representations that completely capture the spatial norm
attainment for a given subalgebra. Classically, the Choquet boundary is the
minimal boundary of a function algebra and it coincides with the collection of
peak points. We investigate the question of minimality for the non-commutative
counterpart of the Choquet boundary and show that minimality is equivalent to
what we call the Bishop property. Not every operator algebra has the Bishop
property, but we exhibit classes of examples that do. Throughout our analysis,
we exploit various non-commutative notions of peak points for an operator
algebra. When specialized to the setting of $C^*$-algebras, our techniques
allow us to provide a new proof of a recent characterization of those
$C^*$-algebras admitting only finite-dimensional irreducible representations.
