Noncommutative integration

Abstract

We will show that if $\mathcal{M}$ is a factor, then for any pair $\varphi ,p\in\mathcal{M}_*^+$ of normal positive linear functionals on $\mathcal{M}$, the inequality: $$ \|\varphi\|\leq \|\psi\| $$ is equivalent to the fact that there exist a countable family $\{\varphi_i : i\in I\}\subset \mathcal{M}_*^+$ in $\mathcal{M}_*^+$ and a family $\{u_i : i\in I\}\subset\mathcal{M}$ of partial isometries in $\mathcal{M}$ such that $$ \varphi=\sum_{ i\in I} \varphi_i,\quad \sum_{ i\in I} u_i{\varphi_i}u_i^*\leq \psi, \quad \text{and} \quad u_i^*u_i=s(\varphi_i), i\in I, $$ where $s(\omega), \omega\in \mathcal{M}_*^+$, means the support projection of $\omega$. Furthermore, if $\|\varphi\|=\|\psi\|$, then the equality replaces the inequality in the second statement. In the case that $\mathcal{M}$ is not of type III$_1$, the family of partial isometries can be replaced by a family of unitaries in $\mathcal{M}$. One cannot expect to have this result in the usual integration theory. To have a similar result, one needs to bring in some kind of non-commutativity. Let $\{X, \mu\}$ be a $\sigma$-finite semifinite measure space and $G$ be an ergodic group of automorphisms of $L^\infty(X,\mu)$, then for a pair $f$ and $g$ of $\mu$-integrable positive functions on $X$, the inequality: $$ \int_X f(x)\text{d} \mu(x)\leq \int_X g(x)\text{d} \mu(x) $$ is equivalent to the existence of a countable families $\{f_i: i\in I\}\subset L^1(X, \mu)$ of positive integrable functions and $\{\gamma_i: i\in I\}$ in $G$ such that $$ f=\sum_{ i\in I} f_i\quad\text{and}\quad \sum_{ i\in I} \gamma_i(f_i)\leq g, $$ where the summation and inequality are all taken in the ordered Banach space $L^1(X, \mu)$ and the action of $G$ on $L^1(X, \mu)$ is defined through the duality between $L^\infty(X, \mu)$ and $L^1(X, \mu)$, i.e., $$ (\gamma(f))(x)=f(\gamma^{-1} x)\frac{d\mu\circ \gamma^{-1}}{d\mu}(x), \quad f\in L^1(X, \mu). $$