Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 161
A linear mapTbetween von Neumann algebras is callednormal
if it is continuous with respect to relevant ultraweak topologies.
Example of a normal map is given by the map of the form
M 3 y7→x∗yx, wherex∈B(H)is such thatx∗yx ∈Mfor each
y∈M. Using the identification from the above paragraph and a
general fact regarding the elements of the Banach space bidual we
see that normal functionals inM∗are precisely those which belong
toM∗. Examples of normal functionals onM⊂B(H)are given by
vector functionals, that is, maps of the formM 3 x7→ 〈ξ,xη〉, where
ξ,η∈H. The uniqueness property of the GNS construction (up to
a unitary equivalence) leads to an easy proof of the following
result.
Proposition 2.5.8 LetAbe aC∗-algebra with a faithful stateω∈
A∗and letα∈Aut(A)be such thatω◦α=ω. Then there exists a
(unique) normal automorphismα ̃∈Aut(πω(A)′′)such that
α ̃(πω(a)) =πω(α(a)), a∈A.Exercise 2.5.1 Prove the last proposition using information
contained in the notes. Can you see the similarities/differences
between this extension ofαand the one appearing in the crossed
product considerations?
It is easy to observe that in the situation described in the last
proposition there exists a (unique) normal stateω ̃ ∈ S(π(A)′′)
such that fora ∈ Athere is ω ̃(πω(a)) = ω(a). Moreover, the
automorphismα ̃preservesω ̃.
Exercise 2.5.2 Prove the above statements.
Generally, ifMis a von Neumann algebra andω ∈S(M)is a
normal state, we will writeα∈Aut(M,ω)to express the fact thatα
is a normal automorphism ofMpreserving the stateω:ω=ω◦α.
Note that the GNS construction can be also applied to von
Neumann algebras. In that case in a sense the construction
changes very little (as the resulting von Neumann algebra is
isomorphic to the original one), but often still remains very useful.
We state some of its aspects in the following proposition.
Proposition 2.5.9 Let M be a von Neumann algebra and
ω∈S(M)a faithful normal state. Then the GNS representationπω