184 From Classical Mechanics to Quantum Field Theory. A Tutorial
to themoment problemin probability measure theory. To adopt this paradigm
we have thus to assume that the set of observables must include at least all real
polynomialsp(A) whenever it contains the observableA. This is in agreement
with the much stronger requirement(AA1).
2.4.1.1 The GNS reconstruction theorem
The set of algebraic states onASis a convex subset in the dualA′SofAS:ifω 1 and
ω 2 are positive and normalized linear functionals,ω=λω 1 +(1−λ)ω 2 is clearly
still the same for anyλ∈[0,1].
Hence, just as we saw for the standard formulation, we can definepure algebraic
statesas extreme elements of the convex body.
Definition 2.4.2.An algebraic stateω:A→Con theC∗-algebra with unitAis
called apure algebraic stateif it is extreme in the set of algebraic states. An
algebraic state that is not pure is calledmixed.
Surprisingly, most of the entire abstract apparatus introduced, given by aC∗-
algebra and a set of states, admits elementary Hilbert space representations when
a reference algebraic state is fixed. This is by virtue of a famous procedure that
Gelfand, Najmark and Segal came up with, and that we prepare to present[26;
27; 25; 5; 6].
Theorem 2.4.3(GNS reconstruction theorem).LetAbe aC∗-algebra with unit
11 andω:A→Ca positive linear functional withω(11) = 1. Then the following
holds.
(a)There exist a triple(Hω,πω,Ψω),whereHωis a Hilbert space, the map
πω:A→B(Hω)aA-representation overHωandΨω∈Hω, such that:(i) Ψωis cyclic forπω.Inotherwords,πω(A)Ψωis dense inHω;
(ii)〈Ψω,πω(a)Ψω〉=ω(a)for everya∈A.(b)If(H,π,Ψ)satisfies (i) and (ii), there exists a unitary operatorU:Hω→
Hsuch thatΨ=UΨωandπ(a)=Uπω(a)U−^1 for anya∈A.Remark 2.4.4.The GNS representationπω:A→B(Hω)is a∗-homomorphism
and thus (c) in remark 2.4.1 applies. In particularπωis norm decreasing and
continuous. Moreover, again referring to the same remark, ifπωis faithful – i.e.,
injective – it is isometric and preserves the spactra of the elements. Ifa∈A
is selfadjointπω(a)is a selfadjointoperatorand its spectrum has the well-known
quantum meaning. This meaning, in view of the property of permanence of the