Stinespring's construction as an adjunction

Abstract

Given a representation of a unital $C^*$-algebra $\mathcal{A}$ on a Hilbert space $\mathcal{H}$, together with a bounded linear map $V:\mathcal{K}\to\mathcal{H}$ from some other Hilbert space, one obtains a completely positive map on $\mathcal{A}$ via restriction using the adjoint action associated to $V$. We show this restriction forms a natural transformation from a functor of $C^*$-algebra representations to a functor of completely positive maps. We exhibit Stinespring's construction as a left adjoint of this restriction. Our Stinespring adjunction provides a universal property associated to minimal Stinespring dilations and morphisms of Stinespring dilations. We use these results to prove the purification postulate for all finite-dimensional $C^*$-algebras.