Mathematical Foundations of Quantum Mechanics 151Remark 2.3.37.It is possible to prove that a von Neumann algebra is always a
direct sum or a direct integral of factors. Therefore factors play a crucial role.
The classification of factors, started by von Neumann and Murray, is one of the
key chapters in the theory of operator algebras, and has enormous consequences in
the algebraic theory of quantum fields. The factors isomorphic toB(H 1 )for some
complex Hilbert spaceH 1 , are called oftypeI. These factors admit atoms, fulfil the
covering property (orthomodularity and irreducibility are always true). Regarding
separability, it depends on separability ofH 1 and requires a finer classification in
factors oftypeInwherenis a cardinal number. There are however factors of type
IIandIIIwhich do not admit atoms and are not important in elementary QM.
The center of the von Neumann algebra of observables enters the physical theory
in a nice way. A common situation dealing with quantum systems is the existence
of amaximal set of compatible observables, i.e. a finite maximal classA=
{A 1 ,...,An}of pairwise compatible observables. The notion of maximality here
means that, if a (bounded) selfadjoint operator commutes with all the observables
inA,thenitisafunctionof them. In perticular it is an observable as well. In view
of proposition 2.3.34 the existence of a maximal set of compatible observables is
equivalent to say that there is a finite set of observablesAsuch thatA′=A′′.We
have the following important consequence
Proposition 2.3.38.If a quantum physical system admits a maximal set of com-
patible observables, then the commutantR′of the von Neumann algebra of observ-
ablesRis Abelian and coincides with the center ofR.
Proof. As the spectral measures of eachA∈Abelong toR,itmustbe(i)A′′⊂R.
SinceA′=A′′,(i)yieldsA′⊂Rand thus, taking the commutant, (ii)A′′⊃R′.
Comparing (i) and (ii) we haveR′⊂R.InotherwordsR′=R′∩R.Inparticular,
R′must be Abelian.
Example 2.3.39.
(1)Considering a quantum particle without spin and referring to the rest space
R^3 of an inertial reference frame,H=L^2 (R^3 ,d^3 x). A maximal set of compatible
observables is the set of the three position operatorsA 1 ={X 1 ,X 2 ,X 3 }or the
the set of the three momenta operatorsA 2 ={P 1 ,P 2 ,P 3 }.Ris the von Neumann
algebra generated byA 1 ∪A 2. It is possible to prove that the commutant (which
coincides with the center) of this von Neumann algebra is trivial (as it includes
a unitary irreducible representation of the Weyl-Heisenberg group) so thatR=
B(H) (see also Theorem 2.3.78).