Linear Radon-Nikodym Theorems for States on a von Neumann Algebra

Abstract

Several linear Radon-Nikodym theorems for states on a von Neumann algebra are obtained in the context of a one parameter family of positive cones introduced by H. Araki. Among other results, we determine when a normal state ϕ admits a linear Radon-Nikodym derivative with respect to a distinguished normal faithful state _ϕ_0 in the sense of Sakai, that is, ϕ = _h ϕ_0 + _ϕ_0 h with a positive h in the algebra.