# nLab Gelfand-Naimark-Segal construction

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

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,

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

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 picture of quantum mechanics: the AQFT and the FQFT picture of quantum field theory. In the former one considers $C^\ast$-algebras of observables, in the latter the spaces of states. Given a $C^\ast$-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)