The Gelfand–Naimark–Segal (GNS) construction establishes a correspondence between cyclic -representations of -algebras and certain linear functionals (usually called states) on those same -algebras. The correspondence comes about from an explicit construction of the *-representation from one of the linear functionals (states).
\rho(x)= \langle \xi, \pi(x)\xi \rangle
for every in .
and this needs to be finished. I don’t have the head for this right now.
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 -algebras of observables, in the lattes spaces of states. Given a -algebra of observables, the corresponding space of state can be taken to be that given by the GNS construction.