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

Abstract

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 f and (W, Qj Ψ (QW) ^ 0 for all QeR where j Ψ (Q) = J Ψ QJ Ψ .The strong closure Vψ of the vectors Qjψ(QW is shown to be a /^-invariant pointed closed convex cone which algebraically span the Hubert space H. Any Jsr-invariant φ e H has a unique decomposition Φ = Φ 1 -Φ 2 such that φ s e Vψ and S*(Φ ί )±8*(Φ*).There exists a unique bijective homeomorphism σ Ψ from the set of all normal linear functionals on R onto Vψ such that the expectation functional by the vector σ Ψ {p) is p.It satisfies ^ {Iky (ft) + tfr(ft) 111 Iky (ft) -*r(ft) II Any two σ Ψ and σψ» are related by a unitary u f in R r by u'σψ{ρ) -σψ'(ρ) for all p.The relation lρ x ^ p 2 holds if and only if there exists AipJpJeR such that A(ft/ft)σ y (ft) = σ Ψ (ρ 2 ).The smallest i is given by 11 AipJpJ 11.It satisfies the chain rule A(pJρ 2 )A(ρ 2 /p 1 ) = A(p 3 lpi).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 j Ψ j φ is an inner automorphism of R. For a product state ® ^ of a C * algebra generated by finite W* tensor products {(8W iyΘIΘie/l/} of von Neumman algebras R jf it is shown that 0pj and ® p$ are equivalent if and only if Σ\ \ a ¥ ( Pj ) -σψ(p£) 11 2 < oo where 11 σ Ψ {p) -σ Ψ {p') \ \ 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)xUψ(g)* = g(x) for all xeR and Uψ(g)σ Ψ (g*p) = σ Ψ )p) for all normal positive linear functionals p.