nLab Gelfand-Naimark-Segal construction



Functional analysis

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



The Gelfand-Naimark-Segal construction (“GNS construction”) represents a state on a star-algebra over the complex numbers – which a priori is defined purely algebraically as a non-degenerate positive linear function

ρ:𝒜, \rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} \,,

– by a vector ψ\psi \in \mathcal{H} in a complex Hilbert space \mathcal{H} as the “expectation value

ρ(A) =ψ|A|ψ ψ,π(A)ψ \begin{aligned} \rho(A) & = \langle \psi \vert \, A \, \vert \psi \rangle \\ & \coloneqq \langle \psi, \pi(A) \psi \rangle \end{aligned}

with respect to some star-representation

π:𝒜End() \pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})

of 𝒜\mathcal{A} on (a dense subspace of) \mathcal{H}; where ,\langle -,-\rangle denotes the inner product on the Hilbert space.

Originally this was considered for C*-algebras and C*-representations (Gelfand-Naimark 43, Segal 47), see for instance (Schmüdgen 90), but the construction applies to general unital star algebras 𝒜\mathcal{A} (Khavkine-Moretti 15) as well as to other coefficient rings, such as to formal power series algebras over [[]]\mathbb{C}[ [\hbar] ] (Bordemann-Waldmann 96).

The GNS-construction plays a central role in algebraic quantum field theory (Haag 96, Moretti 18, Khavkine-Moretti 15), where 𝒜\mathcal{A} plays the role of an algebra of observables and ρ:𝒜\rho \colon \mathcal{A} \to \mathbb{C} the role of an actual state of a physical system (whence the terminology) jointly constituting the “Heisenberg picture”-perspective of quantum physics; so that the GNS-construction serves to re-construct a corresponding Hilbert space of states as in the Schrödinger picture of quantum physics. In this context the version for C*-algebras corresponds to non-perturbative quantum field theory, while the generalization to formal power series algebras corresponds to perturbative quantum field theory.


under construction

For C *C^\ast-algebras



  1. a C*-algebra 𝒜\mathcal{A};

  2. a state ρ:𝒜\rho \;\colon\; \mathcal{A} \to \mathbb{C}

there exists

  1. a C*-representation

    π:𝒜End() \pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})

    of 𝒜\mathcal{A} on some Hilbert space \mathcal{H}

  2. a cyclic vector ψ ρ\psi_\rho \in \mathcal{H}

such that ρ\rho is the state corresponding to ψ ρ\psi_\rho, in that

ρ(A) =ψ ρ|A|ψ ρ ψ ρ,π(A)ψ ρ \begin{aligned} \rho(A) & = \langle \psi_\rho \vert\, A \, \vert \psi_\rho \rangle \\ & \coloneqq \langle \psi_\rho , \pi(A) \psi_\rho \rangle \end{aligned}

for all A𝒜A \in \mathcal{A}.

Namely, consider on the underlying complex vector space of 𝒜\mathcal{A} the sesquilinear form (inner product)

A,B ρρ(A *B). \langle A,B \rangle_\rho \;\coloneqq\; \rho \big( A^\ast B \big) \,.

Since ρ\rho is a “positive functional”, hence taking non-negative values, this is, in general, positive semi-definite. It becomes positive definite on the the quotient vector space

(1)𝒜/N \mathcal{H} \;\coloneqq\; \mathcal{A}/N

by the subspace of 0-norm elements

(2)N{A𝒜|ρ(A *A)=0}. N \;\coloneqq\; \big\{ A \in \mathcal{A} \,\vert\, \rho(A^\ast A) = 0 \big\} \,.

In fact, NN is a left ideal in 𝒜\mathcal{A} so that the left multiplication action of 𝒜\mathcal{A} on itself descends to an action on the quotient Hilbert space (1)

𝒜 π (A,[ψ]) [Aψ]. \array{ \mathcal{A} \otimes \mathcal{H} & \overset{ \;\;\;\;\; \pi \;\;\;\;\; }{ \longrightarrow } & \mathcal{H} \\ (A, [\psi]) &\mapsto& [A \cdot \psi] \,. }

Therefore, the cyclic vector in question can be taken to be that represented by the unit element 1𝒜1 \in \mathcal{A}:

ψ ρ[1]. \psi_\rho \;\coloneqq\; [1] \,.

Hence on this Hilbert space \mathcal{H}, the original operator-algebraic state ρ\rho is now represented by the tautological density matrix

|ψ ρψ ρ=|[1][1]. \left\vert \psi_\rho \right\rangle \left\langle \psi_\rho \right\rangle \;=\; \left\vert [1] \right\rangle \left\langle [1] \right\rangle \,.

For C *C^\ast-categories

The GNS construction for C *C^\ast-algebras is a special case of a more general construction of Ghez, Lima and Roberts applied to C*-categories (horizontal categorification of C *C^\ast-algebras).


Let 𝒞\mathcal{C} be a C *C^\ast-category. Fix an object AOb𝒞A \in \operatorname{Ob}\mathcal{C} and let σ\sigma be a state on the C *C^\ast-algebra 𝒞(A,A)\mathcal{C}(A,A). Then there exists a **-representation

ρ σ:𝒞Hilb \rho_\sigma \colon \mathcal{C} \to \mathbf{Hilb}

together with a cyclic vector ξρ σ(A)\xi \in \rho_\sigma(A) such that for all x𝒞(A,A)x \in \mathcal{C}(A,A),

σ(x)=ξ,ρ σ(x)ξ. \sigma(x) = \langle \xi, \rho_\sigma(x)\xi \rangle.

A C*-algebra 𝒜\mathcal{A} is a C *C^\ast-category with a single object \bullet, where we make the identification A=𝒜(,)A = \mathcal{A}(\bullet,\bullet). In this case the theorem reduces to the classical GNS construction.

Functorial Aspects

See Functorial Aspects of the GNS Representation.

quantum probability theoryobservables and states


The original construction for C*-algebras and C*-representations is due to:

  • Israel Gelfand, Mark Naimark, On the imbedding of normed rings into the ring of operators on a Hilbert space, Matematicheskii Sbornik. 12 (2): 197–217 (1943)

    reprinted in:

    Robert Doran (ed.), C *C^\ast-Algebras: 1943–1993, Contemporary Mathematics 167, AMS 1994 (doi:10.1090/conm/167)

  • Irving Segal, Irreducible representations of operator algebras, Bull. Am. Math. Soc. 53: 73–88, (1947) (pdf, euclid)

Textbook accounts:

in the context of algebraic quantum field theory:

  • Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras, Texts and Monographs in Physics. Springer (1996).

  • Valter Moretti, Spectral Theory and Quantum Mechanics :Mathematical Structure of Quantum Theories, Symmetries and introduction to the Algebraic Formulation, 2nd ed. Springer Verlag, Berlin (2018)

Review with an eye towards quantum probability and entropy:

  • A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, Section II of: Algebraic approach to entanglement and entropy, Phys. Rev. A 88, 022301 (2013) (arXiv:1301.1300)

See also

For general unital star-algebras:

in relation with the classical moment problem and the notion of POVM

  • Nicolò Drago, Valter Moretti, The notion of observable and the moment problem for *\ast-algebras and their GNS representations, Lett. Math. Phys. 2020 (

For formal power series algebras over [[]]\mathbb{C}[ [ \hbar ] ]:

Discussion in terms of universal properties in (higher) category theory is in

Last revised on July 9, 2023 at 08:11:20. See the history of this page for a list of all contributions to it.