Mathematical Foundations of Quantum Mechanics 185spectrum, can be directly attributed to the spectrum ofa∈A:Ifa∈Arepresents
an abstract observable,σ(a)is the set of the possible values attained bya.
As we initially said, it turns out that different algebraic states ω,ω′ gen-
erally give rise to unitarily inequivalent GNS representations (Hω,πω,Ψω)and
(Hω′,πω′,Ψω′): There is no isometric surjective operatorU:Hω′→Hωsuch that
Uπω′(a)U−^1 =πω(a) ∀a∈A.The fact that one may simultaneously deal with all these inequivalent represen-
tations is a representation of the power of the algebraic approach with respect to
the Hilbert space framework.
However one may also focus on states referred to as a fixed GNS representation.
Ifωis an algebraic state onA, every statistical operator on the Hilbert space of a
GNS representation ofω– i.e. every positive, trace-class operator with unit trace
T∈B 1 (Hω) – determines an algebraic state
Aa→tr(Tπω(a)),evidently. This is true, in particular, for Φ∈Hωwith||Φ||ω=1,inwhichcase
the above definition reduces to
Aa→〈Φ,πω(a)Φ〉ω.Definition 2.4.5. Ifωis an algebraic state on theC∗-algebra with unitA, every
algebraic state onAobtained either from a density operator or a unit vector, in
a GNS representation ofω,iscallednormal stateofω.TheirsetFol(ω)is the
foliumof the algebraic stateω.
Note that in order to determineFol(ω) one can use a fixed GNS representation
ofω. In fact, as the GNS representation ofωvaries, normal states do not change,
as implied by part (b) of the GNS theorem.
2.4.1.2 Pure states and irreducible representations
To conclude we would like to explain how pure states are characterised in the
algebraic framework. To this end we have the following simple result (e.g., see[26;
27; 25; 5].
Theorem 2.4.6(Characterization of pure algebraic states).Letωbe an algebraic
state on theC∗-algebra with unitAand(Hω,πω,Ψω)a corresponding GNS triple.
Thenωis pure if and only ifπωis irreducible.