Dilation theory in finite dimensions and matrix convexity

Abstract

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems from their classical infinite-dimensional counterparts. In addition to providing unified proofs of known finite-dimensional dilation theorems, we establish finite-dimensional versions of Agler's theorem on rational dilation on an annulus, of Berger's dilation theorem for operators of numerical radius at most $1$, and of the Putinar-Sandberg numerical range dilation theorem. As a key tool, we prove versions of Carathéodory's and of Minkowski's theorem for matrix convex sets.

27 pages; added more dilation theorems

References in corpus (2)

• Arveson extreme points span free spectrahedra
• Crouzeix's Conjecture and related problems

Cited by in corpus (2)

• Facial structure of matrix convex sets
• Finite-dimensionality in the non-commutative Choquet boundary: peaking phenomena and ${\mathrm{C}}^{\ast }$-liminality