论文标题
关于操作员的平等重量
On the equality of operator valued weights
论文作者
论文摘要
G. K. Pedersen和M. Takeaki在1973年证明，如果$φ$是忠实的，半精灵的，正常的重量弱$ {}^*$ - 密集$σ^φ$ -Invariant $*$ - $ \ mathfrak的subalgebra {m}_φ\; \！$，然后$ψ=φ\; \！$。 1978年，L.Zsidó扩大了上述结果，证明了：如果$φ$如上上述，$ a \ geq 0 $属于centralizer $ m^φ$ $φ\！$ \！$，而$ψ$是$σ^φ$ -INVARIANT，semi-finite，semi-finite，semi-finite，semi-finite on $ m \！ $φ_A：=φ（a^{1/2} \; \！\ cdot \; \！ $ψ=φ_A\; \！$。 在这里，我们将进一步扩展后一个结果，证明了不等式$ψ\leqφ_a$和等价$ψ=φ_a\; \！$的标准。特别关注的是标准，没有$φ$和$ψ\; \！$之间的换向假设，以证明对操作员有价值权重的不平等和平等标准。 关于操作员的价值，证明如果$ e_1 \; \！ ，e_2 $是半填充，普通运算符的重量重量，从von Neumann代数$ m $到von Neumann subalgebra $ n \ ni 1_m $，并且它们在$ \ mathfrak {m} _ {m} _ {e_1}} \; \！; \！; \！$，然后$ e_2 \ e_2 \ leq leq e__1上是平等的。此外，这表明这发生在且仅当任何（或者，如果$ e_1 \; \！，e_2 $）具有同等的支持时，对某些人具有同等的支持。 ，θ\ circ e_1 $在$ \ mathfrak {m} _ {θ\ circ e_1} \; \！$上重合。
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 $σ^φ$-invariant, semi-finite, normal weight on $M\;\!$, equal to $φ$ on the positive part of a weak${}^*$-dense $σ^φ$-invariant $*$-subalgebra of $\mathfrak{M}_φ\;\!$, then $ψ=φ\;\!$. In 1978 L. Zsidó extended the above result by proving: if $φ$ is as above, $a\geq 0$ belongs to the centralizer $M^φ$ of $φ\;\!$, and $ψ$ is a $σ^φ$-invariant, semi-finite, normal weight on $M\;\!$, equal to $φ_a:=φ(a^{1/2}\;\!\cdot\;\! a^{1/2})$ on the positive part of a weak${}^*$-dense $σ^φ$-invariant $*$-subalgebra of $\mathfrak{M}_φ\;\!$, then $ψ=φ_a\;\!$. Here we will further extend this latter result, proving criteria for both the inequality $ψ\leqφ_a$ and the equality $ψ=φ_a\;\!$. 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 $E_1\;\! ,E_2$ 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 $θ$ on $N$ the weights $θ\circ E_2\;\! ,θ\circ E_1$ coincide on $\mathfrak{M}_{θ\circ E_1}\;\!$.
