Boundary representations, geometry of matrix ranges, and C$^*$-envelopes of finite-dimensional operator systems

Abstract

An analysis of the boundary representations and C$^*$-envelopes of some finite-dimensional operator systems $\mathcal R$ is undertaken by considering relationships between operator-theoretic properties of a $d$-tuple $\mathfrak x=(x_1,\dots, x_d)$ of elements in a unital C$^*$-algebra $\mathcal A$ and operator-system properties of the linear span $\mathcal O_{\mathfrak x}$ of $\{1_\mathcal A,x_1,x_1^*,\dots,x_d,x_d^*\}$. This approach lends itself well to the study of certain phenomena in single- and several-variable operator theory, such as the Smith-Ward property. The matrix range of a $d$-tuple of elements $\mathfrak x$ is matrix-affinely homeomorphic to the matrix state space of the operator system determined by $\mathfrak x$, and many of our methods connect the geometry of these compact matrix convex sets to operator system properties, including various forms of nuclearity and lifting properties.