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.
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[2] Luca Giorgetti, Arthur J Parzygnat, Alessio Ranallo, and Benjamin P Russo, "Bayesian inversion and the Tomita–Takesaki modular group", The Quarterly Journal of Mathematics 74 3, 975 (2023).
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