Let φ and ψ be normal
positive linear functionals on a von Neumann algebra M such that ψ ≦ φ. Sakai
proved the existence of a unique element h ∈ M with 0 ≦ h ≦ 1 such that
ψ(x) =φ(hx + xh) for any x ∈ M. A generalization of this theorem is obtained for
weights on von Neumann algebras. Let φ be a faithful normal semi-finite weight and
ψ any weight on M majorized by φ. Then there is a unique element h ∈ M with
0 ≦ h ≦ 1 such that ψ(x) =φ(hx + xh) holds for x in a σ-weakly dense
*-subalgebra of M. A stronger version is obtained when ψ is assumed to be
a normal positive linear functional. Moreover counterexamples are given
to show that in general one can not expect this relation to hold for every
x ∈ M+.
