The noncommutative Choquet boundary III: Operator systems in matrix algebras

We classify operator systems $S\subseteq \mathcal B(H)$ that act on finite dimensional Hilbert spaces by making use of the noncommutative Choquet boundary. S is said to be {\em reduced} when its boundary ideal is 0. In the category of operator systems, that property functions as semisimplicity does in the category of complex Banach algebras. We construct explicit examples of reduced operator systems using sequences of "parameterizing maps" $\Gamma_k: \mathbb C^r\to \mathcal B(H_k)$, $k=1,..., N$. We show that every reduced operator system is isomorphic to one of these, and that two sequences give rise to isomorphic operator systems if and only if they are "unitarily equivalent" parameterizing sequences. Finally, we construct nonreduced operator systems $S$ that have a given boundary ideal $K$ and a given reduced image in $C^*(S)/K$, and show that these constructed examples exhaust the possibilities.