5 Invariant Measures on BMO-states

[page 633, §1]   
[633.1.1] The set of BMO-states B is weak*-compact. [633.1.2] Its open subsets are the elements of the weak*-topology restricted to B. [633.1.3] They generate the σ-algebra B of Borel sets on B. [633.1.4] Let z∈B0⊂B denote an invariant state so that eq. (15) holds for all t∈R,A∈A. [633.1.5] An invariant probability measure on B corresponding to the invariant z can be constructed with the help of a resolution of the identity on B.

[633.2.1] Let Hz,πz,Ωz,Uzt denote the cyclic representation canonically associated with an invariant state z∈B and the time evolution Tt on A. [633.2.2] It is uniquely determined by the two requirements

for A∈A, t∈R and

for t∈R. [633.2.3] Let (,) denotes the scalar product in Hz.

[633.3.1] A resolution of the identity[13, p.301] on the Borel σ-algebra B is a mapping

with the properties

1. P⁢∅=0, P⁢B=1

2. Each P⁢G is a self-adjoint projector.

3. P⁢G∩G′=P⁢G⁢P⁢G′

4. If G∩G′=∅ then P⁢G∪G′=P⁢G+P⁢G′

5. For every ψ∈Hz and ϕ∈Hz the set function Pψ,ϕ:B→C defined by

   Pψ,ϕ⁢G=P⁢G⁢ψ,ϕ

   (29)

   is a complex regular Borel measure on B.

[633.3.2] Because the projectors are self-adjoint the set function Pψ,ψ is a positive measure for every ψ∈Hz. [633.3.3] For ψ=ϕ=Ωz the resulting measure

is an invariant probability measure on the measurable space B,B associated with the invariant BMO-state z∈B. [633.3.4] The triple B,B,Pz is a probability space. [633.3.5] The probability measure Pz is invariant under the adjoint time evolution T*t on B.