Stinespring's construction as an adjunction

Abstract

Given a representation of a C * -algebra together with an isometry, one obtains an operator state on the algebra via restriction using the adjoint action given by the isometry. We show that Stinespring's construction furnishes a left adjoint of this restriction. This is an extension of previous work that showed the Gelfand-Naimark-Segal construction can be viewed as a 2-categorical adjunction. We apply this perspective to analyze Kraus decompositions and properties of the Radon-Nikodym derivative.

Universal Properties of Partial Quantum Maps

“…The role of universal properties and categorical completions in quantum theory, in particular the monoidal indeterminates construction [9], has been studied in recent years [24,14,15,4,10,11] (see also [28] for a related approach in the probabilistic case, and [31,22] for other accounts of dilations as universal properties). The role of partiality in effectus theory was studied in [2].…”