Noncommutative maximal ergodic inequality for non-tracial L1-spaces

TL;DR

This paper extends the noncommutative L1-maximal ergodic inequality from semifinite von Neumann algebras to sigma-finite von Neumann algebras, aiming to facilitate future developments in non-tracial Lp-spaces.

Contribution

It generalizes the noncommutative L1-maximal ergodic inequality to non-tracial settings, broadening the scope of existing results.

Findings

Extended inequality to sigma-finite von Neumann algebras

Bridged gap for non-tracial Lp-space analysis

Provided foundation for future noncommutative ergodic theorems

Abstract

We extend the noncommutative L1-maximal ergodic inequality for semifinite von Neumann algebras established by Yeadon in 1977 to the framework of noncommutative L1-spaces associated with sigma-finite von Neumann algebras. Since the semifnite case of this result is one of the two essential parts in the proof of noncommutative maximal ergodic inequality for tracial Lp-spaces (1<p<infinity) by Junge-Xu in 2007, we hope our result will be helpful to establish a complete noncommutative maximal ergodic inequality for non-tracial Lp-spaces in the future.