On the equality of operator valued weights

G. K. Pedersen and M. Takesaki have proved in 1973 that if \(\varphi\) is a faithful, semi-finite, normal weight on a von Neumann algebra \(M\;\!\), and \(\psi\) is a \(\sigma^{\varphi}\)-invariant, semi-finite, normal weight on \(M\;\!\), equal to \(\varphi\) on the positive part of a weak\({}^*\)-dense \(\sigma^{\varphi}\)-invariant \(*\)-subalgebra of \(\mathfrak{M}_{\varphi}\;\!\), then \(\psi =\varphi\;\!\). In 1978 L. Zsidó extended the above result by proving: if \(\varphi\) is as above, \(a\geq 0\) belongs to the centralizer \(M^{\varphi}\) of \(\varphi\;\!\), and \(\psi\) is a \(\sigma^{\varphi}\)-invariant, semi-finite, normal weight on \(M\;\!\), equal to \(\varphi_a:=\varphi (a^{1/2}\;\!\cdot\;\! a^{1/2})\) on the positive part of a weak\({}^*\)-dense \(\sigma^{\varphi}\)-invariant \(*\)-subalgebra of \(\mathfrak{M}_{\varphi}\;\!\), then \(\psi =\varphi_a\;\!\). Here we will further extend this latter result, proving criteria for both the inequality \(\psi \leq\varphi_a\) and the equality \(\psi =\varphi_a\;\!\). Particular attention is accorded to criteria with no commutation assumption between \(\varphi\) and \(\psi\;\!\), in order to be used to prove inequality and equality criteria for operator valued weights. Concerning operator valued weights, it is proved that if \(E_1\;\! ,E_2\) are semi-finite, normal operator valued weights from a von Neumann algebra \(M\) to a von Neumann subalgebra \(N\ni 1_M\) and they are equal on \(\mathfrak{M}_{E_1}\;\!\), then \(E_2\leq E_1\;\!\). Moreover, it is shown that this happens if and only if for any (or, if \(E_1\;\! ,E_2\) have equal supports, for some) faithful, semi-finite, normal weight \(\theta\) on \(N\) the weights \(\theta\circ E_2\;\! ,\theta\circ E_1\) coincide on \(\mathfrak{M}_{\theta\circ E_1}\;\!\).