Gelfand-Naimark-Segal construction


under construction


The Gelfand–Naimark–Segal (GNS) construction establishes a correspondence between cyclic **-representations of C *C^*-algebras and certain linear functionals (usually called states) on those same C *C^*-algebras. The correspondence comes about from an explicit construction of the *-representation from one of the linear functionals (states).

The nPOV

The GNS construction (as outlined above) is a special case of a more general construction of Ghez, Lima and Roberts applied to C *C^\ast-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.


Proof of Theorem

The Classical Case

A C*C*-algebra AA is a C *C^\ast-category 𝒜\mathcal{A} with one 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.


Given a state, ρ\rho, on some C*-algebra, AA, there is a **-representation π\pi of AA with a cyclic vector ξ\xi whose associated state is ρ\rho. In other words,

ρ(x)=ξ,π(x)ξ \rho(x)= \langle \xi, \pi(x)\xi \rangle

for every xx in AA.


The GNS construction is a central ingredient that translates between the Heisenberg picture and the Schrödinger picture of quantum mechanics: the AQFT and the FQFT picture of quantum field theory. In the former one considers C *C^\ast-algebras of observables, in the latter the spaces of states. Given a C *C^\ast-algebra of observables, the corresponding space of state can be taken to be that given by the GNS construction.

Revised on August 16, 2016 09:19:37 by Todd Trimble (