Dilation theory in finite dimensions and matrix convexity

TL;DR

This paper develops finite-dimensional dilation theorems for operator systems, unifying and extending classical results using matrix convexity, with applications to rational dilation, numerical radius, and numerical range dilation.

Contribution

It provides a unified framework for finite-dimensional dilation theorems derived from classical infinite-dimensional results, including new versions of several key theorems.

Findings

Finite-dimensional Arveson-Stinespring dilation theorem established.

Finite-dimensional versions of Agler's, Berger's, and Putinar-Sandberg theorems proved.

Matrix convexity tools like Carathéodory's and Minkowski's theorems adapted for operator systems.

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 Carath\'{e}odory's and of Minkowski's theorem for matrix convex sets.