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


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.

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\({\mathcal {A}}\) :

Unital \(C^*\)-algebra

\(\mathbf {C}^*\text {-}\mathbf {Alg}\) :

Category of unital \(C^*\)-algebras

\(\omega \) :

A state (on some \(C^*\)-algebra)

\({\mathcal {S}}({\mathcal {A}})\) :

Set of states on \({\mathcal {A}}\)

\(\mathbf {Rep}({\mathcal {A}})\) :

Category of representations of \({\mathcal {A}}\)

\({\mathcal {H}}\) :

Hilbert space

\({\mathcal {B}}({\mathcal {H}})\) :

Bounded linear operators on \({\mathcal {H}}\)

\({\mathcal {S}}\) :

States pre-sheaf

\(\mathbf {States}\) :

States pre-stack

\(\mathbf {Rep}\) :

Representation pre-stack

\(\mathcal {N}_{\omega }\) :

Null-space associated to \(\omega \)


Typical element of \({\mathcal {A}}/\mathcal {N}_{\omega }\)

\({\mathcal {H}}_{\omega }\) :

Hilbert space associated to \(\omega \) via GNS

\(\pi _{\omega }\) :

Representation associated to \(\omega \) via GNS

\(\mathbf {GNS}_{{\mathcal {A}}}\) :

GNS construction for \({\mathcal {A}}\)

\(\mathbf {GNS}_{f}\) :

GNS construction for \({\mathcal {A}}'\xrightarrow {f}{\mathcal {A}}\)

\(\mathbf {GNS}\) :

the GNS construction

\(\Omega \) :

Unit vector (occasionally cyclic)

\((\pi ,{\mathcal {H}},\Omega )\) :

Pointed (or cyclic) representation

\(\mathbf {Rep}^{\bullet }({\mathcal {A}})\) :

Category of pointed representations of \({\mathcal {A}}\)

\(\mathbf {Rep}^{\odot }({\mathcal {A}})\) :

Category of cyclic representations of \({\mathcal {A}}\)

\(\mathbf {rest}_{{\mathcal {A}}}\) :

Restriction to states on \({\mathcal {A}}\) functor

\(\omega _{\Omega }\) :

Vector state \(\langle \Omega ,\;\cdot \;\Omega \rangle \)

\(\mathbf {Rep}^{\bullet }\) :

Pointed representation pre-stack

\(\mathbf {rest}_{f}\) :

Restriction to states for \({\mathcal {A}}'\xrightarrow {f}{\mathcal {A}}\)

\(\mathbf {rest}\) :

Restriction natural transformation

\(\mathbf {GNS}^{\bullet }\) :

Pointed GNS construction

m :

GNS modification


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Correspondence to Arthur J. Parzygnat.

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Communicated by R. Street.

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Parzygnat, A.J. From Observables and States to Hilbert Space and Back: A 2-Categorical Adjunction. Appl Categor Struct 26, 1123–1157 (2018).

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  • States on C*-algebras
  • GNS construction
  • Algebraic quantum theory

Mathematics Subject Classification

  • Primary 81R15
  • Secondary 18D05
  • 46L30