nLab
Gelfand-Naimark-Segal construction

under construction

Idea

The Gelfand–Naimark–Segal (GNS) construction establishes a correspondence between cyclic $*$-representations of $C^{*}$-algebras and certain linear functionals (usually called states) on those same $C^{*}$-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, $\rho$, on some C*-algebra, $A$, there is a $*$-representation $\pi$ of $A$ with a cyclic vector $\xi$ whose associated state is $\rho$. In other words,

$$\rho (x)=⟨\xi ,\pi (x)\xi ⟩$$\rho(x)= \langle \xi, \pi(x)\xi \rangle

for every $x$ in $A$.

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 pircture? of quantum mechnanics?: the AQFT and the FQFT picture of quantum field theory. In the former one considers $C^{*}$-algebras of observables, in the lattes spaces of states. Given a $C^{*}$-algebra of observables, the corresponding space of state can be taken to be that given by the GNS construction.