Vol. 50, No. 2, 1974

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Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule

Huzihiro Araki

Vol. 50 (1974), No. 2, 309–354

For a cyclic and separating vector Ψ of a von Neumann algebra R, the corresponding modular conjugation operator JΨ is characterized by the property that it is an antiunitary involution satisfying JΨΨ = Ψ,JΨRJΨ = Rand ,QjΨ(Q)Ψ) 0 for all Q R where jΨ(Q) = JΨQJΨ.

The strong closure V Ψ of the vectors QjΨ(Qis shown to be a JΨ-invariant pointed closed convex cone which algebraically span the Hilbert space H. Any JΨ-invariant Φ H has a unique decomposition Φ = Φ1 Φ2 such that Φj V Ψ and sR1) sR2).

There exists a unique bijective homeomorphism σΨ from the set of all normal linear functionals on R onto V Ψ such that the expectation functional by the vector σΨ(ρ) is ρ. It satisfies

∥σΨ(ρ1)− σΨ(ρ2)∥2 ≦ ∥ρ1 − ρ2∥
≦ {∥σΨ(ρ1)+ ρΨ(ρ2)∥}∥σ Ψ(ρ1) − σ Ψ(ρ2)∥.

Any two σΨ and σΨ are related by a unitary uin Rby uσΨ(ρ) = σΨ(ρ) for all ρ.

The relation 1 ρ2 holds if and only if there exists A(ρ2∕ρ1) R such that A(ρ2∕ρ1)σΨ(ρ1) = σΨ(ρ2). The smallest l is given by A(ρ2∕ρ1). It satisfies the chain rule A(ρ3∕ρ2)A(ρ2∕ρ1) = A(ρ3∕ρ1). It coincides with the positive square root of the measure theoretical Radon-Nikodym derivative if R is commutative.

As an application, it is shown that product of any two modular conjugation jΨjΦ is an inner automorphism of R.

For a product state ρj of a C algebra generated by finite W tensor products {⊗jIRj}⊗{⊗jI1j} of von Neumman algebras Rj, it is shown that ρj and ρj are equivalent if and only if ΣσΨ(ρj) σΨ(ρj)2 < where σΨ(ρ) σΨ(ρ)is independent of Ψ.

It is shown that there exists a unitary representation UΨ(g) of the group of all -automorphisms of R such that UΨ(g)αjUΨ(g) = g(x) for all x R and UΨ(g)σΨ(gρ) = σΨρ for all normal positive linear functionals ρ.

Mathematical Subject Classification 2000
Primary: 46L10
Received: 29 September 1972
Published: 1 February 1974
Huzihiro Araki