Boundary representations, geometry of matrix ranges, and C${}^{\ast }$-envelopes of finite-dimensional operator systems

An analysis of the boundary representations and C${}^{\ast }$-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${}^{\ast }$-algebra $\mathcal{A}$ and operator-system properties of the linear span ${\mathcal{O}}_{\mathfrak{x}}$ of $\{1_{\mathcal{A}},x_1,x_1^{\ast },\dots ,x_d,x_d^{\ast }\}$. 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.