Non-Commutative Integration

TL;DR

This paper establishes a non-commutative integration framework for factors, characterizing inequalities between positive linear functionals via families of partial isometries or unitaries, and extends classical measure-theoretic results to non-commutative settings.

Contribution

It introduces a novel non-commutative integration theory that generalizes classical measure inequalities using partial isometries and unitaries in von Neumann algebras.

Findings

Characterization of inequalities between positive functionals via partial isometries

Extension of classical measure inequalities to non-commutative algebras

Framework applicable to factors and semifinite measure spaces

Abstract

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}.