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).
The GNS construction (as outlined above) is a special case of a more general construction of Ghez, Lima and Roberts applied to -categories (horizontal categorification of -algebras).
Let be a -category. Fix an object and let be a state on the -algebra . Then there exists a -representation
together with a cyclic vector such that for all ,
A -algebra is a -category with one object , where we make the identification . In this case the theorem reduces to the classical GNS construction.
for every in .
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 -algebras of observables, in the latter the spaces of states. Given a -algebra of observables, the corresponding space of state can be taken to be that given by the GNS construction.