We will show that if $\sM$ is a factor, then for any pair $\f, \p\in\sMdsup$ of normal positive linear functionals on $\sM$, the inequality: $$ \lrnorm{ \f}\leq \lrnorm{ \p} $$ is equivalent to the fact that there exist a countable family $\lrbrace{ \ffdi: i\in I}\subset \sMdsup$ in $\sMdsup$ and a family $\lrbrace{ \udi: i\in I}\i\sM$ of partial isometries in \cM such that $$ \f=\sumd{ i\in I} \ffdi,\quad \sumd{ i\in I} \udi{ \ffdi}\udius\leq \p, \quad \text{ and} \quad \udius\udi=s\lr{ \ffdi}, i\in I, $$ where $s(\omega), \omega\in\sMdsup$, means the support projection of $\omega$. Furthermore, if $\lrnorm{ \f}=\lrnorm{ \p}$, then the equality replaces the inequality in the second statement. In the case that $\sM$ is not of type \threeonec the family of partial isometries can be replaced by a family of unitaries in \cMp One cannot expect to have this result in the usual integration thoery. To have a similar result, one needs to bring in some kind of non-commutativity. Let $\lrbrace{ X, \mu}$ be a $\sig$-finite semifinite measure space and $G$ be an ergodic group of automorphisms of $\linflr{ X, \mu}$, then for a pair $f$ and $g$ of $\mu$-integrable positive functions on $X$, the inequality: $$ \int_X f(x)\txd \mu(x)\leq \int_X g(x)\txd \mu(x) $$ is equivalent to the existence of a countable families $\lrbrace{ \fdi: i\in I}\subset L^1(X, \mu)$ of positive integrable functions and $\lrbrace{ \gdi: i\in I}$ in $G$ such that $$ f=\sumd{ i\in I} \fdi\quad\text{ and}\quad \sumd{ i\in I} \gdi\lr{ \fdi}\leq g, $$ where the summation and inequality are all taken in the oredered Banach space $L^1(X, \mu)$ and the action of $G$ on $\lonelr{ X, \mu}$ is defined through the duality between $\linflr{ X, \mu}$ and $\lonelr{ X, \mu}$, i.e., \lr{ \g(f)}(x)&=f\lr{ \g\inv x}\frac{ \txd\mu\scirc \g\inv}{ \txd\mu}(x), \quad f\in\lonelr{ X, \mu}.
Categories: math.OA math.FA