Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 159
describe the fundamental in quantum dynamics (and more
generally in the theory of operator algebras) Gelfand–Naimark–
Segal construction. It will allow us to build in a natural way
noncommutative ‘measurable’ dynamical systems.
Theorem 2.5.3 (Gelfand, Naimark, Segal) LetAbe aC∗-algebra
and letω∈A∗be a state. Then there exists a Hilbert spaceHω, a
unital representationπω:A→ B(Hω)and a vectorΩ∈Hωsuch
that
Linπω(A)Ω=Hω,ω(a) =〈Ω,πω(a)Ω〉, a∈A.The triple(πω,Hω,Ω)is unique up to a unitary isomorphism: by
this we mean that if(π′,H′,Ω′)is another triple as above, then there
exists a unique unitary operatorU:Hω →H′such thatU(Ω) =
Ω′andπ′(a) =Uπω(a)U∗for alla∈A. Ifωis faithful, then the
representationπω:A→B(Hω)is faithful.
What is the connection of the last result with measurable
dynamical systems? It turns out that a natural class of operator
algebras corresponding to the classical algebras of theL∞-type is
the class ofvon Neumann algebras.
Definition 2.5.4 LetHbe a Hilbert space. For any vectorsξ,η∈H
define a seminormpξ,ηonB(H)by the formula
pξ,η(T) =|〈ξ,Tη〉|, T∈B(H).Theweak operator topologyonB(H)is the (locally convex) topology
determined by the family of seminorms{pξ,η :ξ,η ∈ B(H)}. In
other words, a net(Ti)i∈I converges toTin the weak operator
topology if and only if for all ξ,η ∈ H the net (pξ,η(Ti))i∈I
converges topξ,η(T).
Definition 2.5.5 A von Neumann algebra is a unital∗-subalgebra
ofB(H)closed in the weak operator topology.
von Neumann algebras can be equivalently, and in a sense more
naturally, defined as unital∗-subalgebras closed in theultraweak
(σ-weak) operator topology. The latter, in general finer than the weak
operator topology (but coinciding with it on bounded subsets),
turns out to be the most important (apart from the one coming