For a cyclic and separating
vector Ψ of a von Neumann algebra R, the corresponding modular conjugation
operator J_{Ψ} is characterized by the property that it is an antiunitary involution
satisfying J_{Ψ}Ψ = Ψ,J_{Ψ}RJ_{Ψ} = R′ and (Ψ,Qj_{Ψ}(Q)Ψ) ≧ 0 for all Q ∈ R where
j_{Ψ}(Q) = J_{Ψ}QJ_{Ψ}.
The strong closure V _{Ψ} of the vectors Qj_{Ψ}(Q)Ψ is shown to be a J_{Ψ}invariant
pointed closed convex cone which algebraically span the Hilbert space H. Any
J_{Ψ}invariant Φ ∈ H has a unique decomposition Φ = Φ_{1} − Φ_{2} such that Φ_{j} ∈ V _{Ψ} and
s^{R}(Φ_{1}) ⊥ s^{R}(Φ_{2}).
There exists a unique bijective homeomorphism σ_{Ψ} from the set of all normal
linear functionals on R onto V _{Ψ} such that the expectation functional by the vector
σ_{Ψ}(ρ) is ρ. It satisfies
Any two σ_{Ψ} and σ_{Ψ′} are related by a unitary u′ in R′ by u′σ_{Ψ}(ρ) = σ_{Ψ′}(ρ) for all
ρ.
The relation lρ_{1} ≧ ρ_{2} holds if and only if there exists A(ρ_{2}∕ρ_{1}) ∈ R such that
A(ρ_{2}∕ρ_{1})σ_{Ψ}(ρ_{1}) = σ_{Ψ}(ρ_{2}). The smallest l is given by ∥A(ρ_{2}∕ρ_{1})∥. It satisfies
the chain rule A(ρ_{3}∕ρ_{2})A(ρ_{2}∕ρ_{1}) = A(ρ_{3}∕ρ_{1}). It coincides with the positive
square root of the measure theoretical RadonNikodym derivative if R is
commutative.
As an application, it is shown that product of any two modular conjugation j_{Ψ}j_{Φ}
is an inner automorphism of R.
For a product state ⊗ρ_{j} of a C^{∗} algebra generated by finite W^{∗} tensor products
{⊗_{j∈I}R_{j}}⊗{⊗_{j∈I}1_{j}} of von Neumman algebras R_{j}, it is shown that ⊗ρ_{j} and ⊗ρ_{j}′
are equivalent if and only if Σ∥σ_{Ψ}(ρ_{j}) − σ_{Ψ}(ρ_{j}′)∥^{2} < ∞ where ∥σ_{Ψ}(ρ) − σ_{Ψ}(ρ′)∥ is
independent of Ψ.
It is shown that there exists a unitary representation U_{Ψ}(g) of the group
of all ∗automorphisms of R such that U_{Ψ}(g)αjU_{Ψ}(g)^{∗} = g(x) for all
x ∈ R and U_{Ψ}(g)σ_{Ψ}(g^{∗}ρ) = σ_{Ψ}ρ for all normal positive linear functionals
ρ.
