On the equality of operator valued weights

TL;DR

This paper extends classical results on the equality of weights on von Neumann algebras, providing new criteria for inequalities and equalities of weights and operator valued weights without commutation assumptions.

Contribution

It generalizes previous theorems by establishing new conditions for weight inequalities and equalities, especially for operator valued weights without requiring commutation.

Findings

Criteria for $eta eq eta'$ in weights

Conditions for $ heta eq heta'$ in compositions

Equivalence of weight equality and support conditions

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