On the equality of operator valued weights

Abstract

G. K. Pedersen and M. Takesaki have proved in 1973 that if φ is a faithful, semi-finite, normal weight on a von Neumann algebra M, and ψ is a σφ-invariant, semi-finite, normal weight on M, equal to φ on the positive part of a weak⁎-dense σφ-invariant ⁎-subalgebra of Mφ, then ψ=φ.In 1978 L. Zsidó extended the above result by proving: if φ is as above, a≥0 belongs to the centralizer Mφ of φ, and ψ is a σφ-invariant, semi-finite, normal weight on M, equal to φa:=φ(a1/2⋅a1/2) on the positive part of a weak⁎-dense σφ-invariant ⁎-subalgebra of Mφ, then ψ=φa.Here we will further extend this latter result, proving criteria for both the inequality ψ≤φa and the equality ψ=φa. Particular attention is accorded to criteria with no commutation assumption between φ and ψ, in order to be used to prove inequality and equality criteria for operator valued weights.Concerning operator valued weights, it is proved that if E1, E2 are semi-finite, normal operator valued weights from a von Neumann algebra M to a von Neumann subalgebra N∋1M and they are equal on ME1, then E2≤E1. Moreover, it is shown that this happens if and only if for any (or, if E1, E2 have equal supports, for some) faithful, semi-finite, normal weight θ on N the weights θ∘E2, θ∘E1 coincide on Mθ∘E1.