Strongly peaking representations and compressions of operator systems

TL;DR

This paper explores the use of strongly peaking representations to generalize uniqueness theorems for operator systems, showing that fully compressed systems are uniquely determined by their minimal presentations and generate their C*-envelopes.

Contribution

It introduces a framework connecting strongly peaking representations with minimal presentations, establishing conditions for the uniqueness of separable operator systems.

Findings

Fully compressed separable operator systems generate their C*-envelopes.

Such systems have identities as direct sums of strongly peaking representations.

Under certain conditions, minimal presentations uniquely determine the operator system.

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.