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Given a representation of a C*-algebra, thought of as an abstract collectionof physical observables, together with a unit vector, one obtains a state onthe algebra via restriction. We show that the Gelfand-Naimark-Segal (GNS)construction furnishes a left adjoint of this restriction. To properlyformulate this adjoint, it must be viewed as a weak natural transformation, a1-morphism in a suitable 2-category, rather than as a functor betweencategories. Weak naturality encodes the functoriality and the universalproperty of adjunctions encodes the characterizing features of the GNSconstruction. Mathematical definitions and results are accompanied by physicalinterpretations.

From observables and states to Hilbert space and back: a 2-categorical adjunction