148 From Classical Mechanics to Quantum Field Theory. A Tutorial
At this juncture we are ready to define von Neumann algebras.
Definition 2.3.31.LetHbe a complex Hilbert space. Avon Neumann algebra
inB(H)is a∗-subalgebra ofB(H), with unit, that satisfies any of the equivalent
properties appearing in von Neumann’s theorem 2.3.30.
In particularM′is a von Neumann algebra providedMis a∗-closed subset of
B(H), because (M′)′′=M′as we saw above. Note how, by construction, a von
Neumann algebra inB(H)isaC∗-algebra with unit, or better, aC∗-subalgebra
with unit ofB(H).
It is not hard to see that the intersection of von Neumann algebras is a von
Neumann algebra. IfM⊂B(H) is closed under the adjoint operation,M′′turns
out to be the smallest (set-theoretically) von Neumann algebra containingMas a
subset[ 13 ].ThusM′′is called thevon Neumann algebra generatedbyM.
Since in QM it is natural to deal with unbounded selfadjoint operators, the
definition of commutant is extended to the case of a set of generally unbounded
selfadjoint operators, exploiting the fact that these operators admit spectral mea-
sures made of bounded operators.
Definition 2.3.32.IfNis a set of (generally unbounded) selfadjoint operators in
the complex Hilbert spaceH,thecommutantN′ofN, is defined as the commutant
in the sense of (2.81) of the set of all the spectral measuresP(A)of everyA∈N.
The von Neuman algebraN′′generated byNis defined as(N′)′,wheretheexternal
prime is the one of definition (2.81).
Remark 2.3.33. Notice that, if the selfadjoint operators are all bounded,N′ob-
tained this way coincides with the one already defined in (2.81) as a consequence
of of (ii) and (iv) of Proposition 2.2.70 (for a bounded selfadjoint operatorA).
ThusN′is well-defined and gives rises to a von Neumann algebrabecause the set
of spectral measures is∗-closed. N′′is a von Neumann algebra too for the same
reason.
We are in a position to state a technically important result which concerns both
the spectral theory and the notion of von Neumann algebra[5; 6].
Proposition 2.3.34.LetA={A 1 ,...,An}be a finite collection of selfadjoint
operators in the separable Hilbert spaceHwhose spectral measures commute. The
von Neumann algebraA′′coincides with the collection of operators
f(A 1 ,...,An):=∫
supp(P(A))f(x 1 ,...,xn)dP(A),withf:supp(P(A))→Cmeasurable and bounded.