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.

Complex Free Spectrahedra, Absolute Extreme Points, and Dilations

“…Let A be a tuple of matrices and set C = W(A), K = C 1 . Then [11,Corollary 3.8], λ is a crucial matrix extreme point of W(A) in the sense of [27,Definition 2.4]. This implies that λ is isolated in the extreme points of K. Proposition 3.3 is easily applied to the identification of free spectrahedra, through the polar dual.…”