From Observables and States to Hilbert Space and Back: A 2-Categorical Adjunction

Abstract

Given a representation of a C * -algebra, thought of as an abstract collection of physical observables, together with a unit vector, one obtains a state on the algebra via restriction. We show that the Gelfand-Naimark-Segal (GNS) construction furnishes a left adjoint of this restriction. To properly formulate this adjoint, it must be viewed as a weak natural transformation, a 1-morphism in a suitable 2-category, rather than as a functor between categories. Weak naturality encodes the functoriality and the universal property of adjunctions encodes the characterizing features of the GNS construction. Mathematical definitions and results are accompanied by physical interpretations.

Stinespring's construction as an adjunction

“…Operator state, completely positive, dilation, minimal Stinespring, GNS construction, Stinespring, Radon-Nikodym, quantum operation, algebraic quantum theory. in a certain 2-category of prestacks, category-valued functors, on the category of C *algebras [1].…”

“…In the present work, we do not consider this more general class of completely positive maps. Instead, we focus on the functorial properties of the GNS construction worked out in [1] and extend these results from states to operator states.…”

“…In Section 2, relevant background for completely positive maps is provided. An appropriate category of operator states for a C * -algebra is defined and briefly contrasted with the category of states introduced in [1]. In Section 3, the notion of an anchored representation for a C * -algebra is introduced.…”

“…In Section 3, the notion of an anchored representation for a C * -algebra is introduced. Anchored representations are generalizations of pointed representations described in [1] and a category of anchored representations is introduced. Restricting from anchored representations to operator states is defined in Section 4 and shown to form a natural transformation.…”

“…In the present article, "iff" stands for "if and only if" and is used solely in definitions, C i denotes the i-morphisms of the category (or 2-category) C (when i = 0, these refer to the objects of the category), C * -Alg denotes the category of C * -algebras and unital * -homomorphisms, Cat denotes the 2-category of categories, functors, and natural transformations, and Fun(C * -Alg op , Cat) denotes the 2-category of functors C * -Alg op G G Cat, oplax-natural transformations of such functors, and modifications of oplax-natural transformations. Our previous article on the GNS construction contains further descriptions of these categories including concepts that are not be explicitly defined here [1]. Any claims made without proof are immediate from the definitions.…”