nLab
Gelfand-Naimark-Segal construction

Contents

under construction

Idea

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).

GNS construction

Theorem

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.

and this needs to be finished. I don’t have the head for this right now.

Applications

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 September 12, 2013 17:29:17 by Zoran Škoda (161.53.130.104)