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
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)
Segal, I.E.: Irreducible representations of operator algebras. Bull. Am. Math. Soc. 53, 73–88 (1947)
Mac Lane, S.: Categories for the Working Mathematician, Vol. 5 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1998)
Dixmier, J.: \(C^{*}\)-algebras. North-Holland Publishing Co., Amsterdam. Translated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15. (1977)
Fillmore, P.A.: A User’s Guide to Operator Algebras. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1996)
Hall, B.C.: Quantum Theory for Mathematicians, Vol. 267 of Graduate Texts in Mathematics. Springer, New York (2013)
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)
Hall, B.C.: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, Vol. 222 of Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (2015)
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]
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)
Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, (2008). Reprint of the 1993 edition
Folland, G.B.: Quantum Field Theory: A Tourist Guide for Mathematicians, Vol. 149 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2008)
Bénabou, J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar, pp. 1–77. Springer, Berlin (1967)
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)
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
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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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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DOI: https://doi.org/10.1007/s10485-018-9522-6