On the equality of operator valued weights

Abstract

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. Zsid\'o 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 $\mathfrak{M}_{\theta\circ E_1}\;\!$.