Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 .

Matrix range characterizations of operator system properties

Authors: Benjamin Passer (1), Vern I. Paulsen (2)
Author institution: (1) Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402, U.S.A.
(2) Institute for Quantum Computing and Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

Summary: For finite-dimensional operator systems $\mathcal{S}_{\mathsf T}$, ${\mathsf T} \in B({\mathcal H})^d$, we show that the local lifting property and $1$-exactness of $\mathcal{S}_{\mathsf T}$ may be characterized by measurements of the disparity between the matrix range $\mathcal{W}({\mathsf T})$ and the minimal/maximal matrix convex sets over its individual levels. We then examine these concepts from the point of view of free spectrahedra, direct sums of operator systems, and products of matrix convex sets.

DOI: http://dx.doi.org/10.7900/jot.2019dec16.2278
Keywords: operator system, matrix convex set, matrix range, local lifting property, exactness, Hausdorff distance

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