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.

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Abbreviations

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

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 \(\omega \)

[a]:

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

References

  1. 1.

    Gelfand, I., Neumark, M.: On the imbedding of normed rings into the ring of operators in Hilbert space. Rec. Math. [Mat. Sbornik] N.S. 12(54), 197–213 (1943)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Segal, I.E.: Irreducible representations of operator algebras. Bull. Am. Math. Soc. 53, 73–88 (1947)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Mac Lane, S.: Categories for the Working Mathematician, Vol. 5 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  4. 4.

    Dixmier, J.: \(C^{*}\)-algebras. North-Holland Publishing Co., Amsterdam. Translated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15. (1977)

  5. 5.

    Fillmore, P.A.: A User’s Guide to Operator Algebras. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1996)

    MATH  Google Scholar 

  6. 6.

    Hall, B.C.: Quantum Theory for Mathematicians, Vol. 267 of Graduate Texts in Mathematics. Springer, New York (2013)

    Book  Google Scholar 

  7. 7.

    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Hall, B.C.: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, Vol. 222 of Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (2015)

    Google Scholar 

  9. 9.

    Balachandran, A.P., Govindarajan, T.R., de Quieroz, A.R., Reyes-Lega, A.F.: Entanglement, particle identity and the GNS construction: a unifying approach. Phys. Rev. Lett. 110, 080503 (2013). arXiv:1303.0688 [hep-th]

  10. 10.

    Wald, R.M.: Quantum field theory in curved spacetime and black hole thermodynamics. In: Chicago Lectures in Physics. University of Chicago Press, Chicago, IL (1994)

  11. 11.

    Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, (2008). Reprint of the 1993 edition

    Book  Google Scholar 

  12. 12.

    Folland, G.B.: Quantum Field Theory: A Tourist Guide for Mathematicians, Vol. 149 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2008)

    Book  Google Scholar 

  13. 13.

    Bénabou, J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar, pp. 1–77. Springer, Berlin (1967)

    Google Scholar 

  14. 14.

    Borceux, F.: Handbook of Categorical Algebra. Basic Category Theory, Vol. 50 of Encyclopedia of Mathematics and its Applications, vol. 1. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  15. 15.

    Baez, J.C., Dolan, J.: From finite sets to Feynman diagrams. In: Mathematics Unlimited—2001 and Beyond, pp. 29–50. Springer, Berlin (2001). arXiv:math/0004133

    Google Scholar 

  16. 16.

    Parzygnat, A.: Some 2-Categorical Aspects in Physics. Ph.D. thesis, CUNY Academic Works (2016). http://academicworks.cuny.edu/gc_etds/1475

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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). https://doi.org/10.1007/s10485-018-9522-6

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Keywords

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

Mathematics Subject Classification

  • Primary 81R15
  • Secondary 18D05
  • 46L30