From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 149

2.3.5.2 Lattices of von Neumann algebras

To conclude this elementary mathematical survey, we will say some words about
von Neumann algebras and their associated lattices of orthogonal projectors.
Consider a von Neumann algebraRon the complex Hilbert spaceH.Itis
easy to prove that the setLR(H)⊂Rof the orthogonal projectors included in
Rform a lattice, which is bounded by 0 andI, orthocomplemented with respect
to the orthocomplementation operation of L(H)andσ-complete (because this
notion involves only the strong topology ((iii) in Proposition 2.3.9) andRis closed
with respect to that topology in view of Theorem 2.3.30. MoreoverLR(H)is
orthomodular, and separable like the wholeL(H), assuming thatHis separable.
It is interesting to note that, as expected,LR(H) contains all information about
Ritself since the following result holds.

Proposition 2.3.35.LetRbe a von Neumann algebra on the complex Hilbert
spaceHand consider the latticeLR(H)⊂Rof the orthogonal projectors inR.
then the equalityLR(H)′′=Rholds.

Proof. SinceLR(H)⊂R,wehaveLR(H)′⊃R′andLR(H)′′⊂R′′=R.Let
us prove the other inclusion. A∈Rcan always be decomposed as a linear com-
bination of two selfadjoint operators ofR,A+A†andi(A−A†). So we can
restrict ourselves to the case ofA† =A ∈ R,provingthatA ∈LR(H)′′if
A ∈R.ThePVMofA belongs toRbecause of (ii) and (iv) of Proposition
2.2.70: P(A)commutes with every bounded operatorBwhich commutes with
A.SoP(A)commutes, in particular, with the elements ofR′becauseRA.
We conclude that everyPE(A)∈R′′=R. Finally, there is a sequence of simple
functionssnuniformly converging toidin a compact [−a, a]⊃σ(A)(e.g,see[5;
6 ]). By construction

∫

σ(A)sndP

(A)∈LR(H)′′because it is a linear combination

of elements ofP(A)andLR(H)′′is a linear space. Finally

∫

σ(A)sndP

(A)→Afor

n→+∞uniformly, and thus strongly, as seen in (2) of example 2.2.72. Since
LR(H)′′is closed with respect to the strong topology, we must haveA∈LR(H)′′,
proving thatLR(H)⊃Ras wanted.

2.3.5.3 General algebra of observables and its center

Let us pass to physics and we apply these notions and results. Relaxing the
hypothesis that all selfadjoint operators in the separable Hilbert spaceHassociated
to a quantum system represent observables, there are many reasons to assume that
the observables of a quantum system are represented (in the sense we are going to
illustrate) by the selfadjoint elements of an algebra of von Neumann, we hereafter