Geometry of essential matrix ranges and the Smith-Ward problem for operator systems

Abstract

A $d$-tuple of bounded linear selfadjoint operators acting on an infinite-dimensional separable Hilbert space is said to have the Smith-Ward property if the identity map of the image of the operator system in the Calkin algebra has a completely positive lift. In this paper, we focus on noncommutative geometric properties of a finite dimensional operator system with the goal of understanding how geometric information encoded by the essential matrix range of a spanning set of linear basis for the operator system implies the Smith-Ward property. Some geometric objects of special interest in this paper include one form of noncommutative complex Euclidean ball and maximal noncommutative cubes and polydiscs, as well as some extremal compact matrix convex sets, $K^{\rm min}$ or $K^{\rm max}$, determined by a given compact convex subset $K$ of $\mathbb R^d$.