The Radon–Nikodym theorem for $L^p$-spaces of $W^{\ast }$-algebras

Abstract

We prove an analogue of Sakai's [15] Radon–Nikodym theorem for $L^p$-spaces of $W^{\ast }$-algebras. Following Takesaki [20] we characterize the Radon–Nikodym derivatives by analytic extensions of modular automorphism groups. Finally we relate the facial structure in $L^p$-spaces of $W^{\ast }$-algebras to Connes' [2] unitary Radon–Nikodym cocycles.