  We explore the finite-dimensional part of the non-commutative Choquet
boundary of an operator algebra. In other words, we seek finite-dimensional
boundary representations. Such representations may fail to exist even when the
underlying operator algebra is finite-dimensional. Nevertheless, we exhibit
mechanisms that detect when a given finite-dimensional representation lies in
the Choquet boundary. Broadly speaking, our approach is topological and
requires identifying isolated points in the spectrum of the
$\mathrm{C}^*$-envelope. This is accomplished by analyzing peaking
representations and peaking projections, both of which being non-commutative
versions of the classical notion of a peak point for a function algebra. We
also connect this question with the residual finite-dimensionality of the
$\mathrm{C}^*$-envelope and to a stronger property that we call
$\mathrm{C}^*$-liminality. Recent developments in matrix convexity allow us to
identify a pivotal intermediate property, whereby every matrix state is locally
finite-dimensional.
