The noncommutative Choquet boundary

Let $S$ be an operator system-a self-adjoint linear subspace of a unital $C^*$-algebra $A$ such that $\mathbf 1\in S$ and $A=C^*(S)$ is generated by $S$. A boundary representation for $S$ is an irreducible representation $\pi$ of $C^*(S)$ on a Hilbert space with the property that $\pi\restriction_S$ has a unique completely positive extension to $C^*(S)$. The set $\partial_S$ of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system $S\subseteq C(X)$ that separates points of $X$.

It is known that the closure of the Choquet boundary of a function system $S$ is the Šilov boundary of $X$ relative to $S$. The corresponding noncommutative problem of whether every operator system has ``sufficiently many" boundary representations was formulated in 1969, but has remained unsolved despite progress on related issues. In particular, it was unknown if $\partial_S\neq\emptyset$ for generic $S$. In this paper we show that every separable operator system has sufficiently many boundary representations. Our methods use separability in an essential way.