Strongly peaking representations and compressions of operator systems

Abstract

We use Arveson's notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets which admit minimal presentations. A fully compressed separable operator system necessarily generates the C*-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.

arXiv comment26 pages. Version 2 has minor updates. To appear in International Mathematics Research Notices

References in corpus (12)

• The noncommutative Choquet boundary
• The Choquet boundary of an operator system
• The matricial relaxation of a linear matrix inequality
• Dilations, inclusions of matrix convex sets, and completely positive maps
• Minimal and maximal matrix convex sets
• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization
• Extreme points of matrix convex sets, free spectrahedra and dilation theory
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• Matrix Convex Sets Without Absolute Extreme Points
• Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture
• Compressions of compact tuples
• Shape, Scale, and Minimality of Matrix Ranges

Cited by in corpus (2)

• Empirical properties of optima in free semidefinite programs
• Finite-dimensionality in the non-commutative Choquet boundary: peaking phenomena and ${\mathrm{C}}^{\ast }$-liminality