For normal positive linear functional μ and ν of a
W* algebra \mathfrak{R}, the following extension of a noncommutative Radon-Nikodym theorem by Sakai is given.
There exist decompositions μ=μ
1+μ
2, ν=ν
1+ν
2 such that ν
2 is the smallest normal positive linear functional on \mathfrak{R} satisfying ν{≥}ν
2 and
s(ν
2)⊥s(μ), where
s(α) denotes the support projection of α, and μ
2 is the smallest normal positive linear functional on \mathfrak{R} satisfying μ{≥}μ
2 and
s(μ
2)⊥s(ν). Further, there exists a non-negative self-adjoint operator
A1=
A1(ν/μ) (in general unbounded) such that
A1=∫λdE
λ1 with its spectral projections
Eλ1 in \mathfrak{R}, lim
λ↓0Eλ1=1−
sμν and
ν(s(μ
1)
Qs(μ
1))=μ
1(
A1QA1)≡lim
λ, λ' μ
1(
A1Eλ1QA1Eλ'1)
for all
Q∈\mathfrak{R}, where
sμν=
s(μ
1)−
s(μ
1)∧(1−s(ν)). There also exists another non-negative self-adjoint operator
A2=
A2(ν/μ) such that its spectral projections
Eλ2 are in \mathfrak{R}, lim
λ↓0Eλ2=1−
sνμ for all
Q∈\mathfrak{R},
ν
1(
sνμQsνμ)=μ(
A2QA2).
They are related by
A1(ν/μ)
A2(μ/ν)=
A2(μ/ν)
A1(ν/μ)=
sμν.
The Bures distance function
d(μ, ν) is given by
d(μ/ν)
2=μ(1)+ν(1)−2μ
1(
A1)
=μ(1)+ν(1)−2μ
1(
A2).
In any representation π of \mathfrak{R}, if two vectors Ψ and Φ satisfy ω
Ψ=μ, ω
Φ=ν and ||Ψ−Φ||=
d(μ, ν), where ω
Ψ denotes the vector state by Ψ, then there is a decomposition π=π
1⊕π', Ψ=
x1⊕
x', Φ=
y1⊕
y', ω
x'=μ
2, ω
y'=ν
2,
x1 and
y1 are cyclic vectors of π
1, π
1(
s(μ
1))
y1=π
1(
A1)
x1, π
1(
sνμ)
y1=π
1(
A2)
x1 and such that triplet π
1,
x1 and
y1 are unique up to unitary equivalence for given μ and ν.
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