160 Noncommutative Mathematics for Quantum Systems
from the norm) topology in the study of von Neumann algebras.
More information on the latter topology and the proofs of the
theorems stated below can be found in the monograph [StZ]. Note
that each von Neumann algebra is in particular a (concrete) unital
C∗-algebra.
Theorem 2.5.6 (von Neumann) LetAbe a∗-subalgebra ofB(H)
and letA′ = {T ∈B(H) :∀a∈ATa = aT},A′′ = {T ∈B(H) :
∀a′∈A′Ta′=a′T}. The following are equivalent:
(i)Ais a von Neumann algebra;
(ii)A=A′′.Note that it follows from the above result that ifA is a unital
∗-subalgebra ofB(H)then it is weak-operator dense in its double
commutantA′′. A natural example of a von Neumann algebra,
similar to what we have already seen in Section 2.1, is thevon
Neumann algebra of a discrete groupΓ, VN(Γ). It is defined as the
weak-operator-closure of the reduced groupC∗-algebraCr∗(Γ); in
other words, VN(Γ) =C∗r(Γ)′′⊂B(`^2 (Γ)).
Theorem 2.5.7 (GNS construction – commutative case) LetXbe
a compact space, μ – a regular probability measure on X,
A=C(X). Letω ∈S(A)be given by integration with respect to
the measureμ. Then the GNS construction for the pair(A,ω)leads
to the triple(πμ,L^2 (X,μ), 1X)for which the von Neumann algebra
πμ(C(X))′′ is isometrically isomorphic to the algebra L∞(X,μ)
(the algebra of the essentially bounded functions on (X,μ),
equipped with the essential supremum norm).
Thus, the algebra of the formL∞(X,μ)provides an example of a
(commutative) von Neumann algebra. Its ultraweak operator
topology is the weak∗-topology induced by the duality
L∞(X,μ) = (L^1 (X,μ))∗. In fact this is a general result: Sakai’s
Theorem ([Sa]) characterizes abstract von Neumann algebras as
thoseC∗-algebras that are dual Banach spaces:M= (M∗)∗, where
M∗ is the so-called predual of M(and can be thought of as a
generalizedL^1 -space). Using the standard functional-analytic trick
of embedding a Banach space into its second dual, we often view
the predual ofMas the subspace of the dual ofM:M∗⊂M∗. The
identification ofMwith the dual ofM∗implies in particular that
M∗, viewed as a subspace ofM∗, separates points ofM.